## Chapter 12 – Circles Exercise Ex. 12A

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.

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 PAO, A = 90

By Pythagoras theorem:

Hence, the length of the tangent = 15 cm.

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.

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

In PAO, A = 90,

By Pythagoras theorem:

Hence, the radius of the circle is 7 cm.

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.

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.

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.

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.

OAP = 90

And OBP = 90

So, OAP = OBP = 90

OBP + OAP = (90 + 90) = 180

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

AOBP is a cyclic quadrilateral

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.

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 PCD.

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.

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

PA = PB,

Also, CA = CE and DB = DE

Perimeter of PCD = 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 PCD = 28 cm

A circle is inscribed in a ABC, 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.

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

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

AR, AP are the tangents to the circle

AP = AR = 7cm

AB = 10 cm

BP = AB – AP = (10 – 7)= 3 cm

Also, BP and BQ are tangents to the circle

BP = BQ = 3 cm

Further, CQ and CR are tangents to the circle

CQ = CR = 5cm

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

Hence, BC = 8 cm

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.

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

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

AP + BP + CR + DR = AS + BQ + CQ + DS

(AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)

AB + CD = AD + BC

AD = (AB + CD) – BC = {(6 + 4) – 7} cm = 3 cm

Hence, AD = 3 cm

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.

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

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

In triangle OAP,

Hence, BP = 10.9 cm

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 cm^{2} then find the

lengths of sides AB and AC.

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.

Prove that the line segment joining the

points of contact of two parallel tangents of a circle,

passes through its centre.

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.

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.

## Chapter 12 – Circles Exercise Ex. 12B

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.

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.

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.

In the given figure, common tangents AB and

CD to the two circles with centres O_{1}

and O_{2} intersect at E. Prove that AB = CD.

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.

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 cm^{2} then find the lengths

of sides AB and AC.

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.

Prove that the perpendicular at the point

of contact of the tangent to a circle passes through the centre.

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.

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.

In the given figure, O is the centre of the

circle. PA and PB are tangents. Show that AOBP is a cyclic quadrilateral.

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.

In the given figure, PQ is a chord of a

circle with centre O and PT is a tangent. If ∠QPT = 60°, find ∠PRQ.

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.

## Chapter 12 – Circles Exercise FA

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°

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°

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°

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

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.

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.

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.

Two tangent segments BC and BD are drawn to

a circle with centre O such that ∠CBD = 120°. Prove that OB = 2BC.

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.

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

Prove that the lengths of two tangents

drawn from an external point to a circle are equal.

Prove that the tangents drawn at the ends

of the diameter of a circle are parallel.

In the given figure, if AB = AC, prove that

BE = CE.

If two tangents are drawn to a circle from

an external point, show that they subtend equal angles at the centre.

Prove that the tangents drawn at the ends

of a chord of a circle make equal angles with the chord.

Prove that the parallelogram circumscribing

a circle, is a rhombus.

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.

A quadrilateral is drawn to circumscribe a

circle. Prove that the sums of opposite sides are equal.

Prove that the opposite sides of a

quadrilateral circumscribing a circle subtend supplementary angles at the

centre of the circle.

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.

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.

## Chapter 12 – Circles Exercise MCQ

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

(a) 1

(b) 2

(c) 3

(d) 4

Correct option : (b)

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

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

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 = ?

(a) 30 cm

(b) 28 cm

(c) 25 cm

(d) 18 cm

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

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.

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

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

(a) 8 cm

(b) 10 cm

(c) 12 cm

(d) 16 cm

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

(a) 10 cm

(b) 12 cm

(c) 13 cm

(d) 15 cm

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°

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°

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°

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

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)

(d) 25

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°

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°

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)

(d)

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°

In the given figure, the length of BC is

(a) 7 cm

(b) 10 cm

(c) 14 cm

(d) 15 cm

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

(a) 25°

(b) 45°

(c) 52.5°

(d) 62.5°

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°

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°

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

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°

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

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°

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°

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°

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°

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°

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 cm^{2}

(b) 32.5 cm^{2}

(c) 65 cm^{2}

(d) 30 cm^{2}

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°

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

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°

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

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°

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°

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 cm^{2} then the length of side PQ is

(a) 17.5 cm

(b) 20 cm

(c) 22.5 cm

(d) 25 cm

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

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

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

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°

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

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

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

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

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

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

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

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

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

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.

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.

Which of the following statements is not true?

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

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.

Which of the following statements is not true?

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

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.

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:

- Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
- Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
- Assertion (A) is true and Reason (R) is false.
- 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).

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).

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 | 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).