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R S AGGARWAL AND V AGGARWAL Solutions Mathematics Class 10 Chapter 12 Circles

Chapter 12 – Circles Exercise Ex. 12A

Question 1

Find the length of tangent drawn to a circle with radius 8 cm from a point 17 cm away from the centre of the circle.

Solution 1

PA is the tangent to the circle with center O and radius AO = 8 cm. The point P is at a distance of 17 cm from O.

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

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

By Pythagoras theorem:

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

Hence, the length of the tangent = 15 cm.

Question 2

A point P is 25 cm away from the centre of a circle and the length of tangent drawn from P to the circle is 24 cm. Findthe radius of the circle.

Solution 2

PA is the tangent to the circle with centre O and radius, such that PO = 25 cm, PA = 24 cm

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

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

By Pythagoras theorem:

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

Hence, the radius of the circle is 7 cm.

Question 3

Two concentric circles are of radii 6.5 cm
and 2.5 cm. Find the length of the chord of the larger circle which touches
the smaller circle.

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

Solution 3

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

Question 4

In the given figure, a circle inscribed in
a triangle ABC, touches the sides AB, BC and AC at points D, E and F
respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, find the lengths of
AD, BE and CF.

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

Solution 4

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

Question 5

In the given figure, PA and PB are the tangent segments to a circle with centre O. Show that the points A, O, B and P are concyclic.

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

Solution 5

Given AP is a tangent at A and OA is radius through A and PA and PB are the tangent segments to circle with centre O.

Therefore, OA is perpendicular to AP, similarly, OB is perpendicular to BP.

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

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesOAP = 90

And R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesOBP = 90

So, R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesOAP = R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesOBP = 90

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesOBP + R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesOAP = (90 + 90) = 180

Thus, the sum of opposite angles of quad. AOBP is 180

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesAOBP is a cyclic quadrilateral

Question 6

In the given figure, the chord AB of the
larger of the two concentric circles, with centre O, touches the smaller
circle at C. Prove that AC = CB.

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

Solution 6

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

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

Question 7

From an external point P, tangents PA and PB are drawn to a circle with centre O. if CD is the tangent to the circle at a point E and PA = 14 cm, find the perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesPCD.

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

Solution 7

Given: From an external point P, tangent PA and PB are drawn to a circle with centre O. CD is the tangent to the circle at a point E and PA = 14cm.

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

Since the tangents from an external point are equal, we have

PA = PB,

Also, CA = CE and DB = DE

Perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesPCD = PC + CD + PD

=(PA – CA) + (CE + DE) +(PB – DB)

= (PA – CE) + (CE + DE) + (PB – DE)

= (PA + PB) = 2PA = (2 14) cm

= 28 cm

Hence, Perimeter of R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesPCD = 28 cm

Question 8

A circle is inscribed in a R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesABC, touching AB, BC and AC at P, Q and R respectively. If AB = 10 cm, AR = 7cm and CR = 5 cm, find the length of BC.

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

Solution 8

A circle is inscribed in a triangle ABC touching AB, BC and CA at P, Q and R respectively.

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

Also, AB = 10 cm, AR = 7cm, CR = 5cm

AR, AP are the tangents to the circle

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesAP = AR = 7cm

AB = 10 cm

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesBP = AB – AP = (10 – 7)= 3 cm

Also, BP and BQ are tangents to the circle

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesBP = BQ = 3 cm

Further, CQ and CR are tangents to the circle

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesCQ = CR = 5cm

BC = BQ + CQ = (3 + 5) cm = 8 cm

Hence, BC = 8 cm

Question 9

In the given figure, a circle touches all the four sides of a quadrilateral ABCD whose three sides are AB = 6cm, BC = 7cm and CD = 4 cm. Find AD.

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

Solution 9

Let the circle touches the sides AB, BC, CD and DA at P, Q, R, S respectively

We know that the length of tangents drawn from an exterior point to a circle are equal

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

AP = AS —-(1)    {tangents from A}

BP = BQ —(2)     {tangents from B}

CR = CQ —(3)    {tangents from C}

DR = DS—-(4)    {tangents from D}

Adding (1), (2) and (3) we get

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesAP + BP + CR + DR = AS + BQ + CQ + DS

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Circles(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesAB + CD = AD + BC

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - CirclesAD = (AB + CD) – BC = {(6 + 4) – 7} cm = 3 cm

Hence, AD = 3 cm

Question 10

In the given figure, an isosceles triangle
ABC with AB = AC, circumscribes a circle. Prove that the point of contact P
bisects the base BC.

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

Solution 10

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

Question 11

In the given figure, O is the centre of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circle respectively. If PA = 10 cm, find the length of PB up to one place of decimal

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

Solution 11

Given O is the centre of two concentric circles of radii 4 cm and 6 cm respectively. PA and PB are tangents to the outer and inner circle respectively. PA = 10cm. Join OA, OB and OP.

Then, OB = 4 cm, OA= 6 cm and PA = 10 cm

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

In triangle OAP,

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

Hence, BP = 10.9 cm

Question 12

In the given figure, a triangle ABC is
drawn to circumscribe a circle of radius 3 cm such that the segments BD and
DC into which BC is divided by the point of contact D, are of lengths 6 cm
and 9 cm respectively. If the area of ∆ABC = 54 cm2 then find the
lengths of sides AB and AC.

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

Solution 12

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

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

Question 13

PQ is a chord of length 4.8 cm of a circle
of radius 3 cm. The tangents at P and Q intersect at a point T as shown in
the figure. Find the length of TP.

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

Solution 13

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

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

Question 14

Prove that the line segment joining the
points of contact of two parallel tangents of a circle,
passes through its centre.

Solution 14

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

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

Question 15

In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it
touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If
AB = 29 cm, AD = 23 cm,
B = 90° and DS = 5 cm then find the radius
of the circle.

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

Solution 15

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

Question 16

In the given figure, O is the centre of the
circle and TP is the tangent to the circle from an external point T. If
PBT = 30°, prove that BA: AT = 2:1.

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

Solution 16

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

Chapter 12 – Circles Exercise Ex. 12B

Question 1

In the adjoining figure, a circle touches
all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9
cm and CD = 8 cm. Find the length of side AD.

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

Solution 1

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

Question 2

In the given figure, PA and PB are two
tangents to the circle with centre O. If
APB = 50° then what is the measure of OAB.

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

Solution 2

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

Question 3

In the given figure, O is the centre of a
circle. PT and PQ are tangents to the circle from an external point P. If
TPQ = 70°, find TRQ.

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

Solution 3

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

Question 4

In the given figure, common tangents AB and
CD to the two circles with centres O1
and O2 intersect at E. Prove that AB = CD.

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

Solution 4

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

Question 5

If PT is a tangent to a circle with centre O
and PQ is a chord of the circle such that
QPT = 70°, then find the measure of POQ.

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

Solution 5

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

Question 6

In the given figure, a triangle ABC is
drawn to circumscribe a circle of radius 2 cm such that the segments BD and
DC into which BC is divided by the point of contact D, are of lengths 4 cm
and 3 cm respectively. If the area of ∆ABC = 21 cm2 then find the lengths
of sides AB and AC.

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

Solution 6

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

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

Question 7

Two concentric circles are of radii 5 cm
and 3 cm. Find the length of the chord of the larger circle (in cm) which
touches the smaller circle.

Solution 7

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

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

Question 8

Prove that the perpendicular at the point
of contact of the tangent to a circle passes through the centre.

Solution 8

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

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

Question 9

In the given figure, two tangents RQ and RP
are drawn from an external point R to the circle with centre O. If
PRQ =120°, then prove that OR = PR + RQ.

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

Solution 9

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

Question 10

In the given figure, a circle inscribed in
a triangle ABC touches the sides AB, BC and CA at points D, E and F
respectively. If AB = 14 cm, BC = 8 cm and CA = 12 cm. Find the lengths AD,
BE and CF.

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

Solution 10

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

Question 11

In the given figure, O is the centre of the
circle. PA and PB are tangents. Show that AOBP is a cyclic quadrilateral.

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

Solution 11

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

Question 12

In two concentric circles, a chord of
length 8 cm of the larger circle touches the smaller circle. If the radius of
the larger circle is 5 cm then find the radius of the smaller circle.

Solution 12

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

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

Question 13

In the given figure, PQ is a chord of a
circle with centre O and PT is a tangent. If
QPT = 60°, find PRQ.

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

Solution 13

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

Question 14

In the given figure, PA and PB are two
tangents to the circle with centre O. If
APB = 60° then find the measure of OAB.

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

Solution 14

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

Chapter 12 – Circles Exercise FA

Question 1

In the given figure, O is the centre of a circle, PQ is
a chord and the tangent PT at P makes an angle of 50° with PQ. Then,
POQ = ?

(a) 130°

(b) 100°

(c) 90°

(d) 75°

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

Solution 1

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

Question 2

If the angle between two radii of a circle
is 130° then the angle between the tangents at the ends of the radii is

(a) 65°

(b) 40°

(c) 50°

(d) 90°

Solution 2

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

Question 3

If tangents PA and PB from a point P to a
circle with centre O are drawn so that
APS = 80° then POA = ?

(a) 40°

(b) 50°

(c) 80°

(d) 60°

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

Solution 3

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

Question 4

In the given figure, AD and AE are the
tangents to a circle with centre O and BC touches the circle at F. If AE = 5
cm then perimeter of  ∆ABC is

(a) 15 cm

(b) 10 cm

(c) 22.5 cm

(d) 20 cm

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

Solution 4

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

Question 5

In the given figure, a quadrilateral ABCD
is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch
the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm
and AS = 5 cm, find x.

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

Solution 5

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

Question 6

In the given figure, PA and PB are the
tangents to a circle with centre O. Show that the points A, O, B, P are
concyclic.

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

Solution 6

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

Question 7

In the given figure, PA and PB are two
tangents from an external point P to a circle with centre O. If
PBA = 65°, find OAB and APB.

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

Solution 7

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

Question 8

Two tangent segments BC and BD are drawn to
a circle with centre O such that
CBD = 120°. Prove that OB = 2BC.

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

Solution 8

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

Question 9

Fill in the blanks.

i. A line intersecting a circle in two distinct points is called a ______ .

ii. A circle can have ________  parallel tangents at the most.

iii. The common point of a tangent to a circle and the circle is called the ________ .

iv. A circle can have _________ tangents.

Solution 9
  1. A line intersecting a circle in two distinct points is called a secant.
  2. A circle can have two parallel tangents at the most.
  3. This is since we can draw only parallel tangents on either side of a diameter.
  4. The common point of a tangent to a circle and the circle is called the point of contact.
  5. A circle can have infinitely many tangents.

Question 10

Prove that the lengths of two tangents
drawn from an external point to a circle are equal.

Solution 10

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

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

Question 11

Prove that the tangents drawn at the ends
of the diameter of a circle are parallel.

Solution 11

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

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

Question 12

In the given figure, if AB = AC, prove that
BE = CE.

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

Solution 12

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

Question 13

If two tangents are drawn to a circle from
an external point, show that they subtend equal angles at the centre.

Solution 13

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

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

Question 14

Prove that the tangents drawn at the ends
of a chord of a circle make equal angles with the chord.

Solution 14

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

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

Question 15

Prove that the parallelogram circumscribing
a circle, is a rhombus.

Solution 15

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

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

Question 16

Two concentric circles are of radii 5 cm
and 3 cm respectively. Find the length of the chord of the larger circle
which touches the smaller circle.

Solution 16

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

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

Question 17

A quadrilateral is drawn to circumscribe a
circle. Prove that the sums of opposite sides are equal.

Solution 17

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

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

Question 18

Prove that the opposite sides of a
quadrilateral circumscribing a circle subtend supplementary angles at the
centre of the circle.

Solution 18

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

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

Question 19

Prove that the angle between the two
tangents drawn from an external point to a circle is supplementary to the
angle subtended by the line segments joining the points of contact at the
centre.

Solution 19

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

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

Question 20

PQ is a chord of length 16 cm of a circle
of radius 10 cm. The tangents at P and Q intersect at a point T as shown in
the figure. Find the length of TP.

Solution 20

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

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

Chapter 12 – Circles Exercise MCQ

Question 1

The number of tangents that can be drawn from an external circle is

(a) 1

(b) 2

(c) 3

(d) 4

Solution 1

Correct option : (b)

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

We can draw only 2 tangents from an external point to a circle.

Question 2

In the given figure, RQ is a tangent to the circle with centre O. If SQ = 6 cm and QR = 4 cm, then OR is equal to

(a) 2.5 cm

(b) 3 cm

(c) 5 cm

(d) 8 cm

Solution 2

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

Question 3

In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT = 24 cm. If 0 is the centre of the circle, then length OP = ?

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

(a) 30 cm

(b) 28 cm

(c) 25 cm

(d) 18 cm

Solution 3

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

Question 4

Which of the following pairs of lines in a circle cannot be parallel?

(a) two chords

(b) a chord and a tangent

(c) two tangents

(d) two diameters

Solution 4

Correct option: (d)

The diameter of the circle always passes through the centre. This means all the diameters of a given circle will intersect at the centre, and hence they cannot be parallel.

Question 5

The chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is

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

Solution 5

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

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

Question 6

In the given figure, PT is a tangent to the circle with centre O. If OT = 6 cm and OP =10 cm, then the length of tangent PT is

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

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) 8 cm

(b) 10 cm

(c) 12 cm

(d) 16 cm

 

Solution 6

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

Question 7

In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then, the radius of the circle is

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

(a) 10 cm

(b) 12 cm

(c) 13 cm

(d) 15 cm

Solution 7

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

Question 8

PQ is a tangent to a circle with centre O at the point P. If ∆OPQ is an isosceles triangle, then OQP is equal to

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Solution 8

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

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

Question 9

In the given figure, AB and AC are tangents to the circle with centre O such that BAC = 40°.

Then, BOC is equal to

(a) 80°

(b) 100°

(c) 120°

(d) 140°

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

Solution 9

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

Question 10

If a chord AB subtends an angle of 60° at the centre of a circle, then the angle between the tangents to the circle drawn from A and B is

(a) 30°

(b) 60°

(c) 90°

(d) 120°

Solution 10

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

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

Question 11

In the given figure, O is the centre of two concentric circles of radii 6 cm and 10 cm. AB is a chord of outer circle which touches the inner circle. The length of chord AB is

(a) 8 cm

(b) 14 cm

(c) 16 cm

(d) 136 cm

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

Solution 11

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

Question 12

In the given figure, AB and AC are tangents to a circle with centre O and radius 8 cm. If OA =17 cm, then the length of AC (in cm) is

(a) 9

(b) 15

(c) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Circles 

(d) 25

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

Solution 12

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

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

Question 13

In the given figure, O is the centre of a circle, AOC is its diameter such that ACB = 50°. If AT is the tangent to the circle at the point A then BAT=?

(a) 40°

(b) 50°

(c) 60°

(d) 65°

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

Solution 13

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

Question 14

In the given figure, O is the centre of a circle, PQ is a chord and PT is the tangent at P. If POQ = 70°, then TPQ is equal to

(a) 35°

(b) 45°

(c) 55°

(d) 70°

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

Solution 14

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

Question 15

In the given figure, AT is a tangent to the circle with centre O such that OT = 4 cm and OTA = 30°. Then, AT= ?

(a) 4 cm

(b) 2 cm

(c) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Circles 

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

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

Solution 15

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

Question 16

If PA and PB are two tangents to a circle with centre O such that AOB =110° then APB is equal to

(a) 55°

(b) 60°

(c) 70°

(d) 90°

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

Solution 16

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

Question 17

In the given figure, the length of BC is

(a) 7 cm

(b) 10 cm

(c) 14 cm

(d) 15 cm

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

Solution 17

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

Question 18

In the given figure, if AOD = 135° then BOC is equal to

(a) 25°

(b) 45°

(c) 52.5°

(d) 62.5°

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

Solution 18

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

Question 19

In the given figure, O is the centre of a circle and PT is the tangent to the circle. If PQ is a chord such that QPT = 50° then POQ = ?

(a) 100°

(b) 90°

(c) 80°

(d) 75°

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

Solution 19

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

Question 20

In the given figure, PA and PB are two tangents to the circle with centre O. If APB = 60° then OAB is

(a) 15°

(b) 30°

(c) 60°

(d) 90°

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

Solution 20

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

Question 21

If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm then the length of each tangent is

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

Solution 21

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

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

Question 22

In the given figure, PQ and PR are tangents to a circle with centre A. If QPA = 27° then QAR equals

(a) 63°

(b) 117°

(c) 126°

(d) 153°

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

Solution 22

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

Question 23

In the given figure, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA PB, then the length of each tangent is

(a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

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

Solution 23

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

Question 24

If PA and PB are two tangents to a circle with centre O such that APB = 80°. Then, AOP=?

(a) 40°

(b) 50°

(c) 60°

(d) 70°

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

Solution 24

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

Question 25

In the given figure, Q is the centre of the circle. AB is the tangent to the circle at the point P. If APQ = 58° then the measure of PQB is

(a) 32°

(b) 58°

(c) 122°

(d) 132°

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

Solution 25

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

Question 26

In the given figure, O is the centre of the circle. AB is the tangent to the circle at the point P. If PAO = 30° then CPB + ACP is equal to

(a) 60°

(b) 90°

(c) 120°

(d) 150°

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

Solution 26

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

Question 27

In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If PAB = 67°, then the measure of AQB is

(a) 73°

(b) 64°

(c) 53°

(d) 44°

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

Solution 27

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

Question 28

In the given figure, two circles touch each other at C and AB is a tangent to both the circles. The measure of ACB is

(a) 45°

(b) 60°

(c) 90°

(d) 120°

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

Solution 28

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

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

Question 29

O is the centre of a circle of radius 5 cm. At a distance of 13 cm from O, a point P is taken. From this point, two tangents PQ and PR are drawn to the circle. Then, the area of quad. PQOR is

(a) 60 cm2

(b) 32.5 cm2

(c) 65 cm2

(d) 30 cm2

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

Solution 29

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

Question 30

In the given figure, PQR is a tangent to the circle at Q, whose centre is O and AB is a chord parallel to PR such that BQR = 70°. Then, AQB = ?

(a) 20°

(b) 35°

(c) 40°

(d) 45°

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

Solution 30

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

Question 31

The length of the tangent from an external point P to a circle of radius 5 cm is 10 cm. The distance of the point from the centre of the circle is

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

Solution 31

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

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

Question 32

In the given figure, O is the centre of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T. If PBO = 30° then PTA =?

(a) 60°

(b) 30°

(c) 15°

(d) 45°

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

Solution 32

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

Question 33

In the given figure, a circle touches the side DF of ∆EDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm then the perimeter of ∆EDF is

(a) 9 cm

(b) 12 cm

(c) 13.5 cm

(d) 18 cm

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

Solution 33

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

Question 34

To draw a pair of tangents to a circle, which are inclined to each other at an angle of 45°, we have to draw tangents at the end points of those two radii, the angle between which is

(a) 105°

(b) 135°

(c) 140°

(d) 145°

Solution 34

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

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

Question 35

In the given figure, O is the centre of a circle; PQL and PRM are the tangents at the points Q and R respectively and S is a point on the circle such that SQL = 50° and SRM = 60°. Then, QSR =?

(a) 40°

(b) 50°

(c) 60°

(d) 70°

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

Solution 35

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

Question 36

In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of ∆PQR = 189 cm2 then the length of side PQ is

(a) 17.5 cm

(b) 20 cm

(c) 22.5 cm

(d) 25 cm

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

Solution 36

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

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

Question 37

In the given figure, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm then the length of QR is (a) 1.9 cm

(b) 3.8 cm

(c) 5.7 cm

(d) 7.6 cm

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

Solution 37

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

Question 38

In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 5 cm, BC =7 cm and CS = 3 cm. Then, the length AB =?

(a) 9 cm

(b) 10 cm

(c) 12 cm

(d) 8 cm

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

Solution 38

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

Question 39

In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 6cm, BP = 5 cm, CQ = 3 cm and DR = 4 cm then perimeter of quad. ABCD is

(a) 18 cm

(b) 27 cm

(c) 36 cm

(d) 32 cm

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

Solution 39

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

Question 40

In the given figure, O is the centre of a circle, AB is a chord and AT is the tangent at A. If AOB =100° then BAT is equal to

(a) 40°

(b) 50°

(c) 90°

(d) 100°

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

Solution 40

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

Question 41

In a right triangle ABC, right-angled at B, BC = 12 cm and AB =5 cm. The radius of the circle inscribed in the triangle is

(a) 1 cm

(b) 2 cm

(c) 3 cm

(d) 4 cm

Solution 41

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

Question 42

In the given figure, a circle is inscribed in a quadrilateral ABCD touching its sides AB, BC, CD and AD at P, Q, R and S respectively. If the radius of the circle is 10 cm, BC = 38 cm, PB = 27 cm and AD CD then the length of CD is

(a) 11 cm

(b) 15 cm

(c) 20 cm

(d) 21 cm

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

Solution 42

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

Question 43

In the given figure, ∆ABC is right-angled at B such that BC= 6 cm and AB = 8 cm. A circle with centre O has been inscribed inside the triangle. OP AB, OQ BC and OR AC. If OP = OQ =OR= x cm then x = ?

(a) 2 cm

(b) 2.5 cm

(c) 3 cm

(d) 3.5 cm

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

Solution 43

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

Question 44

Quadrilateral ABCD is circumscribed to a circle. If AB = 6 cm, BC =7 cm, and CD = 4 cm then the length of AD is

(a) 3 cm

(b) 4 cm

(c) 6 cm

(d) 7 cm

Solution 44

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

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

Question 45

In the given figure, PA and PB are tangents to the given circle such that PA = 5 cm and APB = 60°. The length of chord AB is

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

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

Solution 45

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

Question 46

In the given figure, DE and DF are tangents from an external point D to a circle with centre A. If DE = 5 cm and DE DF then the radius of the circle is

(a) 3 cm

(b) 4 cm

(c) 5 cm

(d) 6 cm

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

Solution 46

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

Question 47

In the given figure, three circles with centres A, B, C respectively touch each other externally. If AB = 5 cm, BC = 7 cm and CA = 6 cm then the radius of the circle with centre A is

(a) 1.5 cm

(b) 2 cm

(c) 2.5 cm

(d) 3 cm

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

Solution 47

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

Question 48

In the given figure, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC = 4 cm then the length of AP is

(a) 15 cm

(b) 10 cm

(c) 9 cm

(d) 7.5 cm

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

Solution 48

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

Question 49

In the given figure, O is the centre of two concentric circles of radii 5 cm and 3 cm. From an external point P tangents PA and PB are drawn to these circles. If PA =12 cm then PB is equal to

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

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

Solution 49

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

Question 50

Which of the following statements is not true?

a. If a point P lies inside a circle, no tangent can be drawn to the circle, passing through P.

b. If a point P lies on the circle, then one and only one tangent can be drawn to the circle at P.

c. If a point P lies outside the circle, then only two tangents can be drawn to the circle from P.

d. A circle can have more than two parallel tangents, parallel to a given line.

Solution 50

Correct option: (d)

Options (a), (b) and (c) are all true.

However, option (d) is false since we can draw only parallel tangents on either side of the diameter, which would be parallel to a given line.

Question 51

Which of the following statements is not true?

  1. A tangent to a circle intersects the circle exactly at one point.
  2. The point common to the circle and its tangent is called the point of contact.
  3. The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact.
  4. A straight line can meet a circle at one point only.

Solution 51

Correct option: (d)

Options (a), (b) and (c) are all true.

However, option (d) is false since a straight line can meet a circle at two points even as shown below.

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

Question 52

Which of the following statements is not true?

  1. A line which intersects a circle in two points, is called a secant of the circle.
  2. A line intersecting a circle at one point only, is called a tangent to the circle.
  3. The point at which a line touches the circle, is called the point of contact.
  4. A tangent to the circle can be drawn from a point inside the circle.
Solution 52

Correct option: (d)

Options (a), (b) and (c) are true.

However, option (d) is false since it is not possible to draw a tangent from a point inside a circle.

Question 53

Assertion-and-Reason

Type Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

  1. Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
  2. Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
  3. Assertion (A) is true and Reason (R) is false.
  4. Assertion (A) is false and Reason (R) is true.

Assertion (A)

Reason (R)

At a point P of a circle with centre O and radius 12 cm, a tangent  PQ of length 16 cm is drawn. Then, OQ = 20 cm.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

The correct answer is (a)/(b)/(c)/(d).

Solution 53

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

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

Question 54

Assertion (A)

Reason (R)

If two tangents are drawn to a circle from an external point then they subtend equal angles at the centre.

A parallelogram circumscribing a circle is a rhombus.

The correct answer is (a)/(b)/(c)/(d).

Solution 54

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

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

Question 55

Assertion (A)

Reason (R)

In the given figure, a quad. ABCD is drawn to circumscribe a given circle, as shown.

Then, AB + BC = AD + DC

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

In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.

The correct answer is (a) / (b) / (c) / (d).

Solution 55

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

 

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