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R S AGGARWAL AND V AGGARWAL Solutions Mathematics Class 10 Chapter 4 Triangles

R S AGGARWAL AND V AGGARWAL Solutions for Class 10 Maths Chapter 4 – Triangles

Chapter 4 – Triangles Exercise Ex. 4A

Question 1

D and E are points on the sides AB and AC respectively of a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC such that DE || BC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(i) If AD = 3.6 cm, AB = 10 cm and AE = 4.5 cm, find EC and AC

(ii) If AB = 13.3 cm, AC = 11.9 cm and EC = 5.1 cm, find AD

(iii) If R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and AC = 6.6 cm, find AE

(iv) If R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and EC = 3.5 cm, find AE

Solution 1

(i) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, DE || BC, AD = 3.6 cm, AB = 10 cm, AE = 4.5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, AC = 12.5 cm and EC = 8cm

      (ii) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, DE || BC, AB = 13.3 cm, AC = 11.9 cm and EC = 5.1 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, AD = 7.7 cm

(iii) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, DE || BC, AC = 6.6 cm, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

?R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

 Hence, AE = 2.4 cm

(iv) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, DE || BC, Given R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence AE = 4 cm

Question 2

D and E are points on the sides AB and AC respectively of a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC such that DE || BC. Find the value of x, when

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(i) AD= x cm, DB= (x – 2) cm, AE = (x + 2) cm and EC = (x – 1) cm

(ii) AD = 4 cm, DB = (x – 4) cm, AE = 8 cm and EC = (3x – 19) cm

(iii) AD = (7x – 4) cm, AE = (5x – 2) cm, DB = (3x + 4) cm and EC = 3x cm

Solution 2

(i) D and E are points on the sides AB and AC respectively of a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC such that DE || BC, AD = x cm, DB = (x – 2) cm,

                AE = (x + 2) cm, EC = (x – 1) cm

                     R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

                Hence, x = 4

(ii) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, DE || BC, AD = 4 cm, DB = (x – 4) cm, AE = 8 cm, EC = (3x – 19) cm

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

  Hence, x = 11

(iii) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, DE || BC, AD = (7x – 4) cm, AE = (5x – 2) cm, DB = (3x + 4)cm, EC = 3x cm

                    R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles           

                           R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 3

D and E are points on the sides AB and AC respectively of a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC. In each of the following cases, determine whether DE || BC or not.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(i) AD = 5.7 cm, DB = 9.5 cm, AE = 4.8 cm and EC = 8 cm

(ii) AB = 11.7 cm, AC = 11.2 cm, BD = 6.5 cm and AE = 4.2 cm

(iii) AB = 10.8 cm, AD = 6.3 cm, AC = 9.6 cm and EC = 4 cm

(iv) AD = 7.2 cm, AE = 6.4 cm, AB = 12 cm and AC = 10 cm

Solution 3

Given: A R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC in which D and E are points on the sides AB and AC respectively.

To prove: DE ||BC

Proof:

(i) AD = 5.7 cm, DB = 9.5 cm, AE = 4.8 cm and EC = 8 cm

    Since D and E are the points on AB and AC respectively.

    R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

    Hence, by the converse of Thales theorem DE || BC

(ii) AB = 11.7 cm, AC = 11.2 cm, BD = 6.5 cm, AE = 4.2 cm

    Since D and E are points on AB and AC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, by the converse of Thales theorem DE is not parallel to BC.

 

           (iii) AB = 10.8 cm, AD = 6.3 cm, AC = 9.6 cm, EC = 4 cm

                Since D and E are the points on AB and AC respectively.

                               R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Therefore, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles (each is equal to 1.4)

Hence by the converse of Thales theorem DE || BC

 

(iv) AD = 7.2 cm, AE = 6.4 cm, AB = 12 cm, AC = 10 cm

     Since D and E are points on the side AB and AC respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, by the converse of Thales theorem DB is not parallel to BC

Question 4

In a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, AD is the bisector of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(i) If AB = 6.4 cm, AC = 8 cm and BD = 5.6 cm, find DC.

(ii) If AB = 10 cm, AC = 14 cm and BC = 6cm, find BD and DC

(iii) If AB = 5.6 cm, BD = 3.2 cm and BC = 6 cm, find AC.

(iv) If AB = 5.6 cm, AC = 4cm and DC = 3 cm, find BC

Solution 4

(i) AB = 6.4 cm, AC = 8 cm, BD = 5.6 cm

            Let BC = x

Now, DC = (BC – BD)

            = (x – 5.6) cm

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, AD is the base for of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA

So, by the angle bisector theorem, We have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, BC = 12.6 cm and DC = (12.6 – 5.6) cm = 7 cm

(ii)  AB = 10 cm, AC = 14 cm, BC = 6cm

     
Let BD = x,

DC = (BC – BD) = (6 – x) cm

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, AD is the bisector of ?R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles?A

So, By angle bisector theorem,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, BD = 2.5 cm and DC = (6 – 2.5) cm = 3.5 cm

 

(iii) AB = 5.6 cm, BD = 3.2 cm and BC = 6 cm

     DC = BC – BD = (6 – 3.2) cm = 2.8 cm

Let AC = x,

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, AD is the bisector of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA

So, by the angle bisector theorem we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, AC = 4.9 cm

(iv) AB = 5.6 cm, AC = 4 cm, DC = 3 cm

Let BD = x,

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, AD is the bisector of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA

So, by the angle bisector theorem, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, BD = 4.2 cm

So BC = BD + AC = (4.2 + 3) cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles BC = 7.2 cm

Question 5

M
is a point on the side BC of a parallelogram ABCD.DM when produced meets AB
product at N. Prove that

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 6

Show that the line segment which joins the midpoints of the oblique sides of a trapezium is parallel to the parallel sides.

Solution 6

Let ABCD be the trapezium and let E and F be the midpoints of AD and BC respectively.

Const: Produce AD and BC to meet at P

                             R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPAB, DC || AB

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 7

In the adjoining figure, ABCD is a trapezium in which CD||AB and its diagonals intersect at O. If AO = (5x – 7) cm, OC = (2x + 1) cm, DO = (7x – 5) cm and OB = (7x + 1) cm, find the value of x.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 7

We know that CD || AB in trap ABCD and its diagonals intersect at O.

Since the diagonals of a trapezium divides each other proportionally therefore, we have

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 8

In
a ∆ABC,
M and N are points on the sides AB and AC respectively such that BM = CN. If ∠B
= ∠C
then show MN ∥
BC.

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBC lie on the same side of BC, as shown in the figure. From a point P on BC, PQ || AB and PR||BD are drawn, meeting AC at Q and CD at R respectively. Prove that QR||AD.

                               R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 9

                   ?R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBC lie on the same side of BC. P is a point on BC, PQ || AB and PR || BD are drawn meeting AC at Q and CD at R respectively.

To Prove: QR || AD

Proof: In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, in R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACD, Q and R the points in AC and CD such that

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQR || AD(by the converse of Thales theorem)

Hence proved.

Question 10

In the given figure, side BC of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is bisected at D and O is any point on AD. BO and CO produced meet AC and AB at E and F respectively, and AD is produced to X so that D is the midpoint of OX. Prove that AO : AX = AF : AB and show that EF||BC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given BD = CD and OD = DX

Join BX and CX

Thus, the diagonals of quad OBXC bisect each other

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOBXC is a parallelogram

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBX || CF and so, OF || BX

Similarly, CX || OE

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABX, OF || BX

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 11

ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles. If PQ produced meets BC at R, prove that R is the midpoint of BC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: ABCD is a parallelogram in which P is the midpoint of DC and Q is a point on AC such thatR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles. PQ produced meets BC at R.

To prove: R is the midpoint of BC

Construction: Join BD

Proof: Since the diagonals of a || gm bisect each other at S such that

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQ is the midpoint of CS

So, PQ || DS.

Therefore, QR || SB.

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCSB, Q is the midpoint of CS and QR || SB.

So R is the midpoint of BC.

Question 12

In the adjoining figure, ABC is a triangle in which AB = AC. If D and E are points on AB and AC respectively such that AD = AE, show that the points B, C, E and D are concyclic.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 12

Given: ABC is a triangle in which AB = AC. D and E are points on AB and AC respectively such that AD = AE

To prove: The points B, C, E and D are concyclic.

Proof: AB = AC (given)

         AD = AE (given)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQuad BCEA is cyclic

Hence, the point B, C, E, D are concyclic

Question 13

In
∆ABC,
the bisector of ∠B
meets AC at D.A line PQ ∥ AC meets AB, BC and BD at P, Q and R
respectively.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Show
that PR ⨯ BQ = QR ⨯ BP.

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Chapter 4 – Triangles Exercise Ex. 4B

Question 1

In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form:

(i)

                     R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(ii)

                         R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iii)

                               R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iv)

                               R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 1

(i) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQ = 50° 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesP = 60° 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesC = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR = 70°

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQPR (by AAA similarity)

(ii) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesEFD

                                R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesD = 70° 

SAS: Similarity condition is not satisfied as R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesD are not included angles.

 

(iii) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCAB R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQRP (SAS Similarity)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iv) In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesEFD and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR

                   FE = 2cm, FD = 3 cm, ED = 2.5 cm

PQ = 4 cm, PR = 6 cm, QR = 5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles FED ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR (SSS similarity)

Question 2

In the given figure, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesODC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOBA, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBOC = 115o and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDO = 70o. Find (i) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDOC (ii) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCO(iii) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOAB(iv) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOBA

                   R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 2

                               R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesODC ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOBC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBOC = 115o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDO = 70o

 

(i) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDOC = (180oR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBOC)

               = (180o – 115o)

               = 65o

(ii)       R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOCD = 180oR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDO – R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDOC

            R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOCD = 180o – (70o + 65o)

                    = 45o

(iii) Now, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABO ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesODC

       R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAOB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCOD (vert. Opp R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless) = 65o

       R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOAB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOCD = 45o

(iv)  R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOBA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesODC(alternate angles) = 70o

      So, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOAB = 45o and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOBA = 70o

Question 3

In the given figure, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOAB ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOCD. If AB = 8 cm, BO = 6.4 cm, OC = 3.5 cm and CD = 5cm, find (i) OA (ii) DO

              R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 3

                    R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOAB R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOCD

          AB = 8 cm, BO = 6.4 cm, CD = 5 cm, OC = 3.5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 4

In the given figure, if R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesB, show that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC. If AD = 3.8 cm, AE = 3.6 cm, BE = 2.1 cm and BC = 4.2 cm, find DE

                        R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 4

                           R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesB, AD = 3.8 cm, AE = 3.6 cm, BE = 2.1 cm, BC = 4.2 cm

Proof:

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA                               (common)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesB                             (given)

Therefore, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC        (AA Criterion)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, DE = 2.8 cm

Question 5

The perimeters of two similar triangles ABC and PQR are 32cm and 24cm respectively. If PQ = 12cm, find AB.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR are similar triangles, therefore corresponding sides of both the triangles are proportional.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, AB = 16 cm

Question 6

The corresponding sides of two similar triangles ABC and DEF are BC = 9.1 cm and EF = 6.5 cm. if the perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF is 25 cm, find the perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF are two similar triangles, therefore corresponding sides of both the triangles are proportional.

Hence, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Let perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = x cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 35 cm

Question 7

In the given figure, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCAB = 90o and AD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC. Show that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBDA ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC. If AC = 75 cm, AB = 1m and BC = 1.25 m, find AD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: AB = 100 cm, BC = 125 cm, AC = 75 cm

Proof:

 In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBDA

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBDA = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesB (common)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBDA(by AA similarities)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, AD = 60 cm

Question 8

In the given figure, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 90o and BD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAC. If AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm, find BC.

   R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 8

      R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given that AB = 5.7 cm, BD = 3.8 cm and CD = 5.4 cm

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCBA and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDB

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCBA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesC = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesC (Common)

Therefore, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCBA R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDB (by AA similarities)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, BC = 8.1 cm

Question 9

In the given figure, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 90o and BD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAC. If BD = 8 cm, AD = 4 cm, find CD.

                             R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 9

                   R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given that BD = 8 cm, AD = 4 cm

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBA and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCB, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBDA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCDB = 90o

  R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCB                                  [each = 90oR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA]

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBA R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCB (by AAA similarity)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, CD = 16 cm

Question 10

P and Q are points on the sides AB and AC respectively of a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC. If AP = 2 cm, PB = 4cm, AQ = 3 cm and QC = 6 cm, show that BC = 3PQ

Solution 10

             R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: P is a point on AB.

Then, AB = AP + PB = (2 + 4) cm = 6 cm

Also Q is a point on AC.

Then, AC = AQ + QC = (3 + 6) cm = 9 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Thus, in R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAPQ and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesA (common)

AndR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAPQ ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC(by SAS similarity)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence proved.

Question 11

ABCD is a parallelogram and E is a point on BC. If the diagonal BD intersect AE at F, prove that AF × FB = EF × FD

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: ABCD is a parallelogram and E is point on BC. Diagonal DB intersects AE at F.

To Prove: AF × FB = EF × FD

Proof: In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAFD and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesEFB

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAFD = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesEFB                         (vertically opposite R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDAF = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBEF                             (Alternate R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence proved.

Question 12

In the given figure, DB R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC, DE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAB and AC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC.

Prove that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 12

              R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In the given figure: DB R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles AB, AC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles BC and DB || AC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

AB is the transversal

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC [Alternate R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless]

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBDE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDBE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles[By AA similarity]

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence proved.

Question 13

A vertical stick of length 7.5m casts a shadow 5 m long on the ground and at the same time a tower casts a shadow 24 m long. Find the height of the tower.

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Let AB be the vertical stick and let AC be its shadow.

Then, AB = 7.5 m and AC = 5 m

Let DE be the vertical tower and let DF be its shadow

Then,DF = 24 m, Let DE = x meters

Now, in R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesEDF,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesEDF by SAS criterion

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, height of the vertical tower is 36 m.

Question 14

In an isosceles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, the base AB is produced both ways in P and Q such that AP × BQ =AC2. Prove that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACP R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBCQ

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 14

       R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACP and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBCQ

CA = CB

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCAB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCBA

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACP R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBCQ

Question 15

In the given figure, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles1 = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles2and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Prove that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCE

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles1 = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles2                                 (given)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles             (given)

Also, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles2 = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles1                         R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, by SAS similarity criterion R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCE

Question 16

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

             R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 16

                           R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: ABCD is a quadrilateral in which AD = BC. P, Q , R, S are the midpoints of AB, AC, CD and BD.

To prove: PQRS is a rhombus

Proof: In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC,

Since P and Q are mid points of AB and AC

Therefore, PQ || BC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles      (Mid-point theorem)

 

Similarly,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesSP || RQ and PQ || SR and PQ = RQ = SP = SR

Hence,PQRS is a rhombus.

Question 17

In
a circle, two chords AB and CD intersect at a point P inside the circle.
Prove that

a. PAC ∼ ∆PDB

b. PA .PB =PC.PD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 18

Two
chords AB and CD of a circle intersect at a point P outside the circle. Prove
that

a. PAC ∼ ∆PDB

b. PA .PB =PC.PD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 19

In
a right triangle ABC, right-angle at B, D is a point on hypotenuse such that
BD ⊥
AC. If DP ⊥
AB and DQ ⊥
BC then prove that

a. DQ2= DP.QC

b. DP2=DQ. AP

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Chapter 4 – Triangles Exercise Ex. 4C

Question 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF and their areas are respectively R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles. If EF = 15.4 cm, find BC

Solution 1

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF,

           area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF = 121R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

        R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

We know that the ratio of the area of two similar triangles is equal to the ratio of the squares of their corresponding sides.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, BC = 11.2 cm

Question 2

The areas of two similar triangles ABC and PQR are in the ratio 9 : 16. If BC = 4.5 cm, find the length of QR.

Solution 2

             R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: ?R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR,

          area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 9 and area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR = 16.

We know that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, QR = 6 cm

Question 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR and ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC) = 4 ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR). If BC = 12 cm, find QR.

Solution 3

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC ~ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR,

         area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 4 area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR.

Let area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR = x. Then area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 4x.

We know that the ratio of the areas of two similar triangle is equal to the ratio of the square of their corresponding sides.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence. QR = 6 cm

Question 4

The areas of two similar triangles are 169R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and 121R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles respectively. If the longest side of the larger triangle is 26 cm, find the longest side of the smaller triangle.

Solution 4

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF such that ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC) = 169R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF) = 121 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

We know that the ratio of the area of similar triangles is equal to the ratio of the square of their corresponding sides.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the longest side of smallest triangle side is 22 cm.

Question 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF and their areas are respectively 100R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and 49 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles. If the altitude of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is 5 cm, find the corresponding altitude of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF 

          ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC) = 100 and ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF) = 49

Let AL and DM be the corresponding altitude of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF respectively such that AL = 5 cm and let DM = x cm

We know that the ratio of the area of two similar triangles is equal to the ratio of the square of corresponding altitudes.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, the required altitude is 3.5 cm

Question 6

The corresponding altitudes of two similar triangles are 6 cm and 9 cm respectively. Find the ratio of their areas.

Solution 6

   R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF

Let AL and DM be the corresponding altitudes of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF respectively such that AL = 6 cm and DM = 9 cm.

We know that the ratio of squares of altitudes of two similar triangles is equal to the ratio of the corresponding areas.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, ratio of their areas = 4 : 9

Question 7

The areas of two similar triangles are 81R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and 49R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other.

Solution 7

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF such that

          ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC) = 81R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF) = 49R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Let AL and DM be the corresponding altitudes of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF respectively, such that AL = 6.3 cm and Let DM = x cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

We know that the ratio of the area of two similar triangles is equal to the ratio of the square of corresponding altitudes:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the required altitude 4.9 cm

Question 8

The areas of two similar triangles are 100R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles and 64 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles respectively. If a median of the smaller triangle is 5.6 cm, find the corresponding median of the other.

Solution 8

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles DEF such that ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC) = 100 cm and ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF) = 64R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Let AP and DQ be the corresponding medians of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDEF respectively such that DQ = 5.6cm.

Let AP = x cm.

We know that the ratio of the areas of two similar triangle is equal be the ratio of the squares of their corresponding medians.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, AP = 7 cm

Question 9

In the given figure, ABC is a triangle and PQ is straight line meeting AB in P and AC in Q. If AP = 1cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm, prove that area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAPQ is R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles of the area of the R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC.

          R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 9

           R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given: AP = 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm

          AB = AP + PB = (1 + 3) cm = 4 cm

          AC = AQ + QC = (1.5 + 4.5) cm = 6 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAPQ and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAPQ = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC                  (corresponding R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAQP = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB                  (corresponding R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAPQ R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC            [by AA similarity]

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence proved.

Question 10

In the given figure, DE||BC. If DE = 3 cm, BC = 6cm and ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE) =R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles, find the area of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Given DE || BC

         DE = 3 cm and BC = 6 cm

         ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE) = 15R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is right angled at A and AD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC. If BC = 13 cm and AC = 5 cm, find the ratio of the areas of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADC, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADC = 90o    (AD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesDCA            (common)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAC R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, the ratio of the areas of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADC = 169:25

Question 12

In the given figure, DE||BC and DE : BC = 3 : 5. Calculate the ratio of the areas of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and the trapezium BCED.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Let DE = 3x and BC = 5x

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC                  (corres. R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAED = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB                  (corres. R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC            (by AA similarity)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Let, ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE) = 9x2 units

Then, ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC) = 25x2 units

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, ratio of ar(R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE) to the ar(trap BCED) = 9:16

Question 13

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, D and E are midpoints of AB and AC respectively. Find the ratio of the areas of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 13

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, D and E are midpoint of AB and AC respectively.

So, DE|| BC and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Now, in R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC                       (corres. R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAED = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACB                       (corres. R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangless)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC                     (by AA similarity)

Let AD = x and AB = 2x

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Therefore, the ratio of the areas of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 1:4

Chapter 4 – Triangles Exercise Ex. 4D

Question 1

The sides of certain triangles are given below. Determine which of them are right triangles:

(i) 9 cm, 16 cm, 18 cm

(ii) 7 cm, 24 cm, 25 cm

(iii) 1.4 cm, 4.8 cm, 5 cm

(iv) 1.6 cm, 3.8 cm, 4 cm

(v) (a-1)cm, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Trianglescm, (a + 1)cm

Solution 1

For a given triangle to be a right angled, the sum of the squares of the two sides must be equal to the square of the largest side.

(i) Let a = 9cm, b = 16 cm and c = 18 cm. Then

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence the given triangle is not right angled.

(ii) Let a = 7cm, b = 24 cm and c = 25 cm, Then

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, the given triangle is a right triangle.

(iii) Let a = 1.4 cm, b = 4.8 cm, and c = 5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, the given triangle is a right triangle

(iv) Let a = 1.6 cm, b = 3.8 cm and c = 4 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Hence, the given triangle is not a right triangle

(v) Let p = (a – 1) cm, q = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Trianglescm and r = (a + 1) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

               Hence, the given triangle is a right triangle

Question 2

Aman goes 80 m due east and then 150 m due north. How far is he from the starting point?

Solution 2

Starting from A, let the man goes from A to B and from B to C, as shown in the figure.

Then,

AB = 80 m, BC = 150 m andR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

From right R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the man is 170m north-east from the starting point.

Question 3

A man goes 10 m due south and then 24 m due west. How far is he from the starting point?

Solution 3

Starting from O, let the man goes from O to A and then A to B as shown in the figure.

Then,

OA = 10 m, AB = 24 m and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesOAB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Using Pythagoras theorem:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the man is 26 m south-west from the starting position.

Question 4

A 13m long ladder reaches a window of a building 12 m above the ground. Determine the distance of the foot of the ladder from the building.

Solution 4

Let AB be the building and CB be the ladder.

Then,

AB = 12 m, CB = 13 m and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCAB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

By Pythagoras theorem, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the distance of the foot of the ladder from the building is 5 m.

Question 5

A ladder is placed in such a way that its foot is at a distance of 15m from a wall and its top reaches a window 20m above the ground. Find the length of the ladder.

Solution 5

Let AB be the wall where window is at B, CB be the ladder and AC be the distance between the foot of the ladder and wall.

Then,

AB = 20 m, AC = 15 m, and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesCAB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

By Pythagoras theorem, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the length of ladder is 25 m.

Question 6

Two vertical poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12m, find the distance between their tops.

Solution 6

Let AB and CD be the given vertical poles.

Then,

AB = 9 m, CD = 14 m and AC = 12 m

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Const: Draw, BE || AC.

Then,

CE = AB = 9m and BE = AC = 12 m

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles DE = (CD – CE)

         = (14 – 9)

         = 5 m

In right R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBED, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, the distance between their tops is 13 m.

Question 7

A
guy wire attached to a vertical pole of height 18 m is 24 m long and has a
stake attached to the other end. How far from the base of the pole should the
stake be drive so that the wire will be taut?

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 8

In the given figure, O is a point inside a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR such that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR = 90o, OP = 6 cm and OR = 8 cm. If PQ = 24 cm and QR = 26 cm, prove that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR is right-angled.

Solution 8

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesQPR = 90o, PQ = 24 cm, and QR = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPOR, PO = 6 cm, QR = 8cm and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPOR = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPOR,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR,

By Pythagoras theorem, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

                       (sum of square of two sides equal to square of greatest side)

Hence, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesPQR is a right triangle which is right angled at P.

Question 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is an isosceles triangle with AB = AC = 13 cm. The length of altitude from A on BC is 5 cm. Find BC.

Solution 9

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is an isosceles triangle with AB = AC = 13 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Const: Draw altitude from A to BC (AL R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC).

Now, AL = 5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesALB,

  R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesALB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesALC,

        R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesALC = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 10

Find the length of altitude AD of an isosceles R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC in which AB = AC = 2a units and BC = a units.

Solution 10

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC in which AB = AC = 2a units and BC = a units

Const: Draw AD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC then D is the midpoint of BC.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is an equilateral triangle of side 2a units. Find each of its altitudes.

Solution 11

In an equilateral triangle all sides are equal.

Then, AB = BC = AC = 2a units

Const: Draw an altitude AD R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC

Given BC = 2a. Then, BD = a

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABD,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADB = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, length of each altitude is R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 12

Find the height of an equilateral triangle of side 12 cm.

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC is an equilateral triangle in which all side are equa.

Therefore, AB = BC = AC = 12 cm

If BC = 12 cm

Then, BD = DC = 6 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADB,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence the height of the triangle is R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 13

Find the length of a diagonal of a rectangle whose adjacent sides are 30 cm and 16 cm.

Solution 13

Let ABCD is the given rectangle, let BD is a diagonal making a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADB.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBAD = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Using Pythagoras theorem:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, length of diagonal DB is 34 cm.

Question 14

Find the length of each side of a rhombus whose diagonals are 24 cm and 10 cm long.

Solution 14

Let ABCD be the given rhombus whose diagonals intersect at O.

Then AC = 24 cm and BD = 10 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

We know that the diagonals of a rhombus bisect each other at right angles.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

From right R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAOB, we have

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence, each side of a rhombus 13 cm

Question 15

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, D is the midpoint of BC and AE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles BC. If AC > AB, show that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 15

Given: R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC in which D is the midpoint of BC. AE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles BC and AC > AB.

Then, BD = CD and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAED = 90o,

Then, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADE < 90o and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesADC > 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAED,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Putting value of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Trianglesfrom (1) in (2), we get

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 16

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC, AB = AC. Side BC is produced to D. Prove that

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Const: Draw a perpendicular AE from A

Thus, AE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles BC

Proof:

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABC,AB = AC

And AE is a bisector of BC

Then,BE = EC

In right angle triangles AED and ACE

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Hence proved.

Question 17

In the given figure, D is the midpoint of side BC and AE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(i)R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(ii)R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iii)R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iv)R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 17

Given: D is the midpoint of side BC, AE R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesBC, BC = a, AC = b, AB = c, ED = x, AD = p and AE = h

In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAEC, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAEC = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(i)             In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAEC, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesAEC = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(ii)In R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABE, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABE = 90o

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iii)Adding (1) and (2), we get

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(iv)Subtracting (2) from (1), we get

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 18

In
∆ABC
,AB = AC. Side BC is produced to D. Prove that

(AD2-AC2)=BD.
CD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 19

ABC is an isosceles triangle, right-angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesABE and R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesACD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 19

ABC is an isosceles triangle right angled at B,

Let AB = BC = x cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

By Pythagoras theorem,

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 21

In
a ∆ABC,
AD is a median and AL ⊥ BC. Prove that:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 22

Naman is doing fly-fishing in a
stream. The tip of this fishing rod is 1.8 above the surface of the water and
the fly at the end of the string rests on the water 3.6 m away from him and
2.4 m from the point directly under the tip of the rod. Assuming that the
string (from the tip of his rod to the fly) is taut, how much string does he
have out) see the adjoining figure)? If he pulls in the string at the rate of
5 cm per second, what will be the; horizontal distance of the fly from him
after 12 seconds?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Chapter 4 – Triangles Exercise Ex. 4E

Question 1

State
the two properties which are necessary for given two triangles to be similar.

Solution 1

Two triangles are said to be similar to each other if:

(i) their corresponding angles are equal, and

(ii)
their corresponding sides are proportional.

Question 2

State
the basic proportionality theorem.

Solution 2

If a line is drawn parallel to one side of a triangle to intersect
the other two sides in distinct point, then the other sides are divided in
the same ratio.

Question 3

State
the converse of ‘Thales’ theorem.

Solution 3

If a line divides any two sides of a triangle in the same ratio then
the line must be parallel to the third side.

Question 4

State
the midpoint theorem.

Solution 4

The line segment joining the mid-points of any two sides of a
triangle is parallel to the third side.

Question 5

State
the AAA-similarity criterion.

Solution 5

If in any two triangles, the corresponding angles are equal, then
their corresponding sides are proportional and hence the triangles are
similar.

Question 6

State
the AA-similarity criterion.

Solution 6

If two angles of one triangle are respectively equal to two angles of
another triangle then the two triangles are similar.

Question 7

State
the SSS-criterion for similarity of triangles.

Solution 7

If the corresponding sides of two triangles are proportional then
their corresponding angles are equal, and hence the two triangles are
similar.

Question 8

State
the SAS-similarity criterion.

Solution 8

If one angle of a triangle is equal to one angle of the other
triangle and the sides including these angles are proportional then the two
triangles are similar.

Question 9

State
Pythagoras’ theorem.

Solution 9

In a right triangle, the square of the hypotenuse is equal to the sum
of the squares of the other two sides.

Question 10

State
the converse of Pythagoras’ theorem.

Solution 10

In a triangle, if the square of one side is equal to the sum of the
squares of the other two sides then the angle opposite to the first side is a
right angle.

Question 11

If
D,E and F are respectively the midpoints of sides
AB,BC and CA of ∆ABC then what is
the ratio of the areas of ∆DEF and ∆ABC?

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 12

Two
triangle ABC and PQR are such that AB =3 cm, AC =6 cm, ∠A
=700 ,PR =9 cm, ∠P = 700 and
PQ = 4.5 cm. Show that ∆ABC ∼ ∆PQR
and state the similarity criterion.

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 13

If
∆ABC

∆DEF
such that 2 AB = DE and BC = 6 cm, find EF.

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 14

In
the given figure, DE ∥ BC such that AD =
x cm , DB
=(3x+4) cm, AE = (x+3) cm and EC = (3x +19) cm. Find the value of x.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 15

A
ladder 10 m long reaches the window of a house 8 m above the ground. Find the
distance of the foot of the ladder from the base of the wall.

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 16

Find
the length of the altitude of an equilateral triangle of side 2a
cm.

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 17

∆ABC ∼

DEF such that ar (∆ABC) =64 cm2
and or (∆DEF)
= 169 cm2. If BC = 4 cm , find EF.

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 18

In
a trapezium ABCD, it is given that AB ∥ CD and AB = 2 CD.Its diagonals AC and BD intersect at the point O such
that ar (∆AOB) = 84 cm2.
Find ar (∆COD)

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 19

The
corresponding sides of two similar triangles are in the ratio 2:3. If the
area of the smaller triangle is 48 cm2, find the area of the
larger triangle.

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 21

Find
the length of each side of a rhombus whose diagonals are 24 cm and 10 cm
long.

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 22

Two
triangles DEF and GHK are such that ∠D = 480
and ∠H
= 570. If ∆DEF ∼ ∆ GHK then find the measure of ∠F.

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 24

In
triangles BMP and CNR it is given that PB = 5 cm, MP = 6 cm, BM = 9 cm and NR
= 9 cm. If ∆BMP ∼ ∆CNR
then find the perimeter of ∆CNR.

Solution 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 25

Each
of the equal sides of an isosceles triangle is 25 cm. Find the length of it
altitude if the base is 14 cm.

Solution 25

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 26

A
man goes 12 m due south and then 35 m due west. How far is he from the
starting point?

Solution 26

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 27

If
the lengths of the sides BC, CA and AB of a ∆ABC
are a ,
b and c respectively and AD is the bisector of ∠A then find the
lengths of BD and DC.

Solution 27

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 28

In
the given figure, ∠AMN = ∠MBC
= 760. If a,b
and c are the lengths of AM, MB and BC respectively then express the length
of MN in terms of a,b and c.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 28

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 29

The
lengths of the diagonals of a rhombus are 40 cm and 42 cm. Find the length of
each side of the rhombus.

Solution 29

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 30

For
each of the following statements state whether true (T) or false (F):

Two
circles with different radii are similar.

Solution 30

Similar figures have the same shape but need not have the same size.

Since all circles irrespective of the radii will have the same shape,
all will be similar.

So, the statement is true.

Question 31

For
each of the following statements state whether true (T) or false (F):

Any
two rectangles are similar.

Solution 31

Two rectangles are similar if their corresponding sides are
proportional.

So, the statement is false.

Question 32

For
each of the following statements state whether true (T) or false (F):

If
two triangles are similar then their corresponding angles are equal and their
corresponding sides are equal.

Solution 32

Two triangles are said to be similar to each other if:

(i) their corresponding angles are equal, and

(ii) their corresponding sides are
proportional.

So, the statement is false.

Question 33

For
each of the following statements state whether true (T) or false (F):

The
length of the line segment joining the midpoints of any two sides of a
triangle is equal to half the length of the third side.

Solution 33

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 34

For
each of the following statements state whether true (T) or false (F):

In
a ∆ABC,
AB = 6 cm, ∠A
= 45° and AC = 8 cm and in a ∆DEF, DF = 9 cm, ∠D
= 45° and DE = 12 cm, then ∆ABC ∼ ∆DEF.

Solution 34

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 35

For
each of the following statements state whether true (T) or false (F):

The
polygon formed by joining the midpoints of the sides of a quadrilateral is a
rhombus.

Solution 35

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

The line segments joining the midpoints of the adjacent sides of a
quadrilateral form a parallelogram as shown.

It may or may not be a rhombus.

So, the statement is false.

Question 36

For
each of the following statements state whether true (T) or false (F):

The
ratio of the areas of two similar triangles is equal to the ratio of their
corresponding angle-bisector segments.

Solution 36

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 37

For
each of the following statements state whether true (T) or false (F):

The
ratio of the perimeters of two similar triangles is the same as the ratio of
their corresponding medians.

Solution 37

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 38

For
each of the following statements state whether true (T) or false (F):

If
O is any point inside a rectangle ABCD then OA2 + OC2 =
OB2 + OD2.

Solution 38

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 39

For
each of the following statements state whether true (T) or false (F):

The
sum of the squares on the sides of a rhombus is equal to the sum of the
squares on its diagonals.

Solution 39

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - TrianglesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Chapter 4 – Triangles Exercise FA

Question 1

∆ABC ∆DEF and their
perimeters are 32 cm and 24 cm respectively. If AB = 10 cm then DE = ?

(a) 8 cm

(b) 7.5 cm

(c) 15 cm

(d) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 2

In the given figure, DE BC. If DE = 5 cm, BC
= 8 cm and AD = 3.5 cm then AB =?

(a) 5.6 cm

(b) 4.8 cm

(c) 5.2 cm

(d) 6.4 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 3

Two poles of height 6 m
and 11 m stand vertically upright on a plane ground. If the distance between
their feet is 12 m then the distance between their tops is

(a) 12 m

(b) 13 m

(c) 14 m

(d) 15 m

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 4

The areas of two similar
triangles are 25 cm2 and 36 cm2 respectively. If the
altitude of the first triangle is 3.5 cm then the corresponding altitude of
the other triangle is

(a) 5.6 cm

(b) 6.3 cm

(c) 4.2 cm

(d) 7 cm

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 5

If ∆ABC ∆DEF such that 2AB =
DE and BC = 6 cm, find EF.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 6

In the given figure, DE BC such that AD = x
cm, DB = (3x+4) cm, AE = (x+3) cm and EC = (3x+19) cm. Find the value of x.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 7

A 10 m long reaches the
window of a house 8 m above the ground. Find the distance of the foot of the
ladder from the base of the wall.

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 8

Find the length of the
altitude of an equilateral triangle of side 2a cm.

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 9

∆ABC ∆DEF such that
ar(∆ABC)=64 cm2 and ar(∆DEF)=169 cm2. If BC=4 cm, find
EF.

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 10

In a trapezium ABCD, it is given that AB CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(∆AOB)=84 cm2. Find ar(∆COD).

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

 

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 11

The corresponding sides
of two similar triangles are in the ratio 2 : 3. If the area of the smaller
triangle is 48 cm2, find the area of the larger triangle.

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 13

Prove that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 15

Find the length of each
side of a rhombus whose diagonals are 24 cm and 10 cm long.

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 16

Prove that the ratio of
the perimeters of two similar triangles is the same as the ratio of their
corresponding sides.

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 19

In the given figure,
∆ABC is an obtuse triangle, obtuse-angled at B. If AD
CB (produced) prove
that AC2=AB2+BC2+2BC.BD.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Chapter 4 – Triangles Exercise MCQ

Question 1

A man goes 24 m due west and then 10 m due north. How far is he from the starting point?

(a) 34 m

(b) 17 m

(c) 26 m

(d) 28 m

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 2

Two poles of height 13 in and 7 m respectively stand vertically on a plane ground at a distance of 8 m from each other. The distance between their tops is

(a) 9 m

(b) 10 m

(c) 11 m

(d) 12 m

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 3

A vertical stick 1.8 m long casts a shadow 45 cm long on the ground. At the same time, what is the length of the shadow of a pole 6 m high?

(a) 2.4 in

(b) 1.35 m

(c) 1.5 m

(d) 13.5 m

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 4

A vertical pole 6 m long casts a shadow of length 3.6 m on the ground. What is the height of a tower which casts a shadow of length 18 m at the same time?

(a) 10.8 m

(b) 28.8 m

(c) 324 m

(d) 30 m

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 5

The shadow of a 5-m-long stick is 2 m long. At the same time the length of the shadow of a 12.5-m-high tree (in m) is   

(a) 3.0

(b) 3.5

(c) 4.5

(d) 5.0

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 6

A ladder 25 m long just reaches the top of a building 24 m high from the ground. What is the distance of the foot of the ladder from the building?

(a) 7 m

(b) 14 m

(c) 21 m

(d) 24.5 m

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

 

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 7

In the given figure, O is a point inside a ∆MNP such that MOP = 90°, OM = 16 cm and OP = 12 cm. If MN = 21 cm and NMP = 90° then NP = ?

(a) 25 cm

(b) 29 cm

(c) 33 cm

(d) 35 an

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 8

The hypotenuse of a right triangle is 25 cm. The other two sides are such that one is 5 an longer than the other. The lengths of these sides are

(a) 10 cm, 15 cm

(b) 15 cm, 20 cm

(c) 12 cm, 17 cm

(d) 13 cm, 18 cm

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 9

The height of an equilateral triangle having each side 12cm, is

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 10

∆ABC is an isosceles triangle with AB =AC = 13 cm and the length of altitude from A on BC is 5 cm. Then, BC=?

(a) 12 cm

(b) 16 cm

(c) 18 cm

(d) 24 cm

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 11

In a ∆ABC it is given that AB = 6 cm, AC = 8 cm and AD is the bisector of A. Then, BD: DC  = ?

(a) 3 : 4

(b) 9 : 16

(c) 4 : 3

(d) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 12

In a ∆ABC it is given that AD is the internal bisector of A. If BD=4cm, DC=5cm and AB=6 cm, then AC=?

(a) 4.5 cm

(b) 8 cm

(c) 9 cm

(d) 7.5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 13

In a ∆ABC, it is given that AD is the internal bisector of A. If AB = 10 cm, AC = 14 cm and BC = 6 cm, then CD = ?

(a) 4.8 cm

(b) 3.5 cm

(c) 7 cm

(d) 10.5 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 14

In a triangle, the perpendicular from the vertex to the base bisects the base. The triangle is

(a) right-angled

(b) isosceles

(c) scalene

(d) obtuse-angled

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 15

In an equilateral triangle ABC, if AD ⊥  BC then which of the following is true?

(a) 2AB2=3AD2

(b) 4AB2=3AD2

(c) 3AB2=4AD2

(d) 3AB2=2AD2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 16

In a rhombus of side 10 cm, one of the diagonals is 12 cm long. The length of the second diagonal is

(a) 20 cm

(b) 18 cm

(c) 16 cm

(d) 22 cm

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 17

The lengths of the diagonals of a rhombus are 24 cm and 10 cm. The length of each side of the rhombus is

(a) 12 cm

(b) 13 cm

(c) 14 cm

(d) 17 cm

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 18

If the diagonals of a quadrilateral divide each other proportionally then it is a

(a) parallelogram

(b) trapezium

(c) rectangle

(d) square

Solution 18

Correct option: (b)

Recall that the diagonals of a trapezium divide each other proportionally.

Note that this happens even in a parallelogram, square and rectangle, but without additional information it is not possible to be sure.

Question 19

In the given figure, ABCD is a trapezium whose diagonals AC and BD intersect at O such that OA=(3x-1) cm, OB=(2x+1)cm, OC=(5x-3)cm and OD=(6x-5)cm. Then, x=?

(a) 2

(b) 3

(c) 2.5

(d) 4 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 20

The line segments joining the midpoints of the adjacent sides of a quadrilateral form

(a) a parallelogram

(b) a rectangle

(c) a square

(d) a rhombus

Solution 20

Correct option: (a)

The line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram as shown below. 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 21

If the bisector of an angle of a triangle bisects the opposite side them the triangle is

(a) scalene

(b) equilateral

(c) isosceles

(d) right-angled

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) 30°

(b) 40°

(c) 45°

(d) 50°

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 23

In ∆ABC, DE BC so that AD = 2.4 cm, AE= 3.2 cm and EC = 4.8 cm. Then, AB =?

(a) 3.6 cm

(b) 6 cm

(c) 6.4 cm

(d) 7.2 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 24

In a ∆ABC, if DE is drawn parallel to BC, cutting AB and AC at D and E respectively such that AB = 7.2 cm, AC = 6.4 cm and AD = 4.5 cm. Then, AE =?

(a) 5.4 cm

(b) 4 cm

(c) 3.6 cm

(d) 3.2 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles  

Question 25

In ∆ABC, DE BC so that AD = (7x-4) cm, AE = (5x-2) cm, DB = (3x+4) cm and EC = 3x cm. Then, we have

(a) x = 3

(b) x = 5

(c) x = 4

(d) x = 2.5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 25

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 26

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) 4.2 cm

(b) 3.1 cm

(c) 2.8 cm

(d) 2.1 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 26

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 27

∆ABC ∆DEF and the perimeters of ∆ABC and ∆DEF are 30 cm and 18 cm respectively. If BC = 9 cm then EF=?

(a) 6.3 cm

(b) 5.4 cm

(c) 7.2 cm

(d) 4.5 cm

Solution 27

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 28

∆ABC ∆DEF such that AB = 9.1 cm and DE = 6.5 cm. If the perimeter of ∆DEF is 25cm, what is the perimeter of ∆ABC?

(a) 35 cm

(b) 28 cm

(c) 42 cm

(d) 40 cm

Solution 28

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 29

In ∆ABC, it is given that AB=9 cm, BC = 6cm and CA = 7.5 cm. Also, ∆DEF is given such that EF = 8cm and ∆DEF ∆ABC. Then, perimeter of ∆DEF is

(a) 22.5 cm

(b) 25 cm

(c) 27 cm

(d) 30 cm

Solution 29

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 30

ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of the areas of triangles ABC and BDE is

(a) 1 : 2

(b) 2 : 1

(c) 1 : 4

(d) 4 : 1

Solution 30

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 31

It is given that ∆ABC ∆DFE. If A = 30°, C=50°, AB = 5cm, AC = 8 cm and DF = 7.5 cm then which of the following is true?

(a) DE = 12 cm, F = 50°

(b) DE = 12 cm, F = 100°

(c) EF = 12 cm, D = 100°

(d) EF = 12 cm, D = 30°

Solution 31

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 32

In the given figure, BAC = 90° and AD BC. Then,

(a) BC·CD = BC2

(b) AB·AC = BC2

(c) BD·CD = AD2

(d) AB·AC = AD2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 32

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 33

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) 45°

(b) 60°

(c) 90°

(d) 120°

Solution 33

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 34

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) B = E

(b) A = D

(c) B = D

(d) A = F

Solution 34

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 35

In ∆DEF and ∆PQR, it is given that D = Q and R = E, then which of the following is not true?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 35

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 36

If ∆ABC ∆EDF and ∆ABC is not similar to ∆DEF then which of the following is not true?

(a) BC·EF = AC·FD

(b) AB·EF = AC·DE

(c) BC·DE = AB·EF

(d) BC·DE = AB·FD

Solution 36

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 37

In ∆ABC and ∆DEF, it is given that B = E, F = C and AB = 3DE, then the two triangles are

(a) congruent but not similar

(b) similar but not congruent

(c) neither congruent nor similar

(d) similar as well as congruent

Solution 37

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 38

In the given figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, APB = 50° and CDP=30° then PBA=?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) 50°

(b) 30°

(c) 60°

(d) 100°

Solution 38

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 39

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) ∆PQR ∆CAB

(b) ∆PQR ∆ABC

(c) ∆CBA ∆PQR

(d) ∆BCA ∆PQR

Solution 39

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 40

Corresponding sides of two similar triangles are in the ratio 4 : 9. Areas of these triangles are in the ratio

(a) 2 : 3

(b) 4 : 9

(c) 9 : 4

(d) 16 : 81

Solution 40

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 41

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 41

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 42

In an equilateral ∆ABC, D is the midpoint of AB and E is the midpoint of AC. Then, ar(∆ABC) : ar(∆ADE)=

?

(a) 2 : 1

(b) 4 : 1

(c) 1 : 2

(d) 1 : 4

Solution 42

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 43

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) 5 : 7

(b) 25 : 49

(c) 49 : 25

(d) 125 : 343

Solution 43

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 44

∆ABC ∆DEF such that ar(∆ABC) = 36 cm2 and ar(∆DEF)=49 cm2. Then, the ratio of their corresponding sides is

(a) 36 : 49

(b) 6 : 7

(c) 7 : 6

(d) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 44

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 45

Two isosceles triangles have their corresponding angles equal and their areas are in the ratio 25:36. The ratio of their corresponding heights is

(a) 25 : 36

(b) 36 : 25

(c) 5 : 6

(d) 6 : 5

Solution 45

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles 

Question 46

The line segments joining the midpoints of the sides of a triangle form four triangles, each of which is

(a) congruent to the original triangle

(b) similar to the original triangle

(c) an isosceles triangle

(d) an equilateral triangle

Solution 46

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Correct option: (b)

The line segments joining the midpoints of the sides of a triangle form four triangles, each of which is similar to the original triangle.

Question 47

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(a) 8 cm

(b) 10 cm

(c) 12 cm

(d) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 47

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 48

In the given figure, O is the point of intersection of two chords AB and CD such that OB = OD and AOC=45°. Then, ∆OAC and ∆ODB are

(a) equilateral and similar

(b) equilateral but not similar

(c) isosceles and similar

(d) isosceles but not similar

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Solution 48

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 49

In an isosceles ∆ABC, if AC = BC and AB2 = 2AC2 then C =?

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Solution 49

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 50

In ∆ABC, if AB = 16 cm, BC = 12 cm and AC = 20 cm, then ∆ABC is

(a) acute-angled

(b) right-angled

(c) obtuse-angled

(d) not possible

Solution 50

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 51

Which of the following is a true statement?

(a) Two similar triangles are always congruent.

(b) Two figures are similar if they have the same shape and size.

(c) Two triangles are similar if their corresponding sides are proportional.

(d) Two polygons are similar if their corresponding sides are proportional.

Solution 51

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 52

Which of the following is a false statement?

(a) If the areas of two similar triangles are equal then the triangles are congruent.

(b) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides.

(c) The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding medians.

(d) The ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding altitudes.

Solution 52

Correct option: (b)

Clearly, option (b) is false since the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

Question 53

Match the following columns

Column I

Column II

(a)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

p

6

(b)

If ∆ABC ∆DEF such that 2AB = 3DE and BC = 6 cm then EF = ……. cm.

q

4

(c)

If ∆ABC ∆PQR such that ar(∆ABC) : ar(∆PQR) = 9 : 16 and BC = 4.5 cm then QR = …….. cm

r

3

(d)

In the given figure, AB CD and OA = (2x+4) cm, OB = (9x-21) cm, OC = (2x-1) cm and OD = 3 cm. Then x =?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

s

2.1

Solution 53

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

Question 54

Match the following columns

Column I

Column II

(a)

A man goes 10m due east and then 20 m due north. His distance from the starting point is ……..m.

p

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(b)

In an equilateral triangle with each side 10 cm, the altitude is ………cm.

q

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(c)

The area of an equilateral triangle having each side 10 cm is …….. cm2.

r

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

(d)

The length of diagonal of a rectangle having length 8 m and breadth 6 m is …… m.

s

10

Solution 54

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Triangles

 

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