Circles

CBSE Class 10 — Maths — Previous Year Questions

93 questions202 total marksVSA: 24MCQ: 26Assertion-Reason: 5Case-Study: 3SA: 25LA: 10

Q1.VSA1 mark(2020)Easy

△ A BC

is circumscribing a circle. Find the length of

BC

.

$Triangle ABC circumscribing a circle with BP=3cm, AP=4cm, AR=11cm$

Q2.VSA1 mark(2020)Easy

In given Fig. 2, the length

PB =

________ cm.

$Two concentric circles with tangent from external point$

Q3.VSA1 mark(2020)Easy

The distance between two parallel tangents of a circle of radius 4 cm is ________.

Q4.VSA1 mark(2020)Easy

In Fig. 2,

P A

and

PB

are tangents to the circle with centre

O

such that

∠ A PB = 50°

, then the measure of

∠ O A B

is ________.

$Circle with tangents PA and PB from external point P$

Q5.VSA1 mark(2020)Medium

In Fig. 3,

PQ

is a chord of a circle and

PT

is tangent at

P

such that

∠ QPT = 60°

, then the measure of

∠ PRQ

is ________.

$Circle with chord PQ and tangent PT at P$

Q6.MCQ1 mark(2020)Easy

In Figure-2,

TP

and

TQ

are tangents drawn to the circle with centre

O

. If

∠ POQ = 115°

then

∠ PTQ

is

$Circle with tangents TP and TQ, angle POQ = 115°$

Q7.MCQ1 mark(2020)Easy

From an external point

Q

, the length of the tangent to a circle is 5 cm and the distance of

Q

from the centre is 8 cm. The radius of the circle is

Q8.VSA1 mark(2020)Easy

In Figure-4,

A B

and

C D

are common tangents to circles which touch each other at

D

. If

A B = 8

cm, then find the length of

C D

.

$Two circles touching at D with common tangents$

Q9.MCQ1 mark(2020)Easy

In Figure 2,

PQ

is tangent to the circle with centre at

O

, at the point

B

. If

∠ A OB = 100°

, then

∠ A BP

is equal to

$Circle with tangent PQ at B, centre O, angle AOB = 100 degrees$

Q10.MCQ1 mark(2020)Easy

In Figure 3, from an external point

P

, two tangents

PQ

and

PR

are drawn to a circle of radius

4

cm with centre

O

. If

∠ QPR = 90°

, then length of

PQ

is

$Two tangents PQ and PR from external point P to circle with centre O, radius 4 cm$

Q11.MCQ1 mark(2020)Easy

All concentric circles are ________ to each other.

Q12.MCQ1 mark(2023)Medium

In the given figure,

PQ

is tangent to the circle centred at

O

. If

∠ A OB = 95°

, then the measure of

∠ A BQ

will be

$Circle with centre O, tangent PQ at B, angle AOB = 95 degrees$

(A)

47.5°

(B)

42.5°

(C)

85°

(D)

95°

Q13.MCQ1 mark(2023)Medium

In the given figure, the quadrilateral PQRS circumscribes a circle. Here

P A + CS

is equal to:

$Quadrilateral PQRS circumscribing a circle$

(a)

QR

(b)

PR

(c)

PS

(d)

PQ

Q14.Assertion-Reason1 mark(2023)Medium

Assertion (A): A tangent to a circle is perpendicular to the radius through the point of contact.
Reason (R): The lengths of tangents drawn from an external point to a circle are equal.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.

Q15.MCQ1 mark(2023)Easy

In the given figure, PQ is a tangent to the circle with centre O. If

∠ OPQ = x

∠ POQ = y

, then

x + y

is:

$Circle with tangent PQ and centre O$

Q16.MCQ1 mark(2023)Easy

In the given figure, TA is a tangent to the circle with centre O such that

OT = 4

cm,

∠ OT A = 30°

, then length of TA is:

$Circle with tangent TA, OT = 4 cm, angle 30 degrees$

Q17.MCQ1 mark(2023)Medium

In the given figure,

A B = BC = 10

cm. If

A C = 7

cm, then the length of

BP

is:

$Triangle with inscribed circle$

Q18.MCQ1 mark(2023)Easy

In the given figure,

A B

is a tangent to the circle centered at

O

. If

O A = 6

cm and

∠ O A B = 30°

, then the radius of the circle is:

$Circle with tangent AB$

Q19.MCQ1 mark(2023)Medium

In the given figure,

A C

and

A B

are tangents to a circle centered at

O

. If

∠ CO D = 120°

, then

∠ B A O

is equal to:

$Circle with tangents AC and AB$

Q20.Assertion-Reason1 mark(2023)Medium

**Assertion (A):** If the points

A (4, 3)

and

B (x, 5)

lie on a circle with centre

O (2, 3)

, then the value of

x

2

.

**Reason (R):** Centre of a circle is the mid-point of each chord of the circle.

Q21.MCQ1 mark(2023)Medium

In the given figure,

P A

and

PB

are tangents from external point

P

to a circle with centre

C

and

Q

is any point on the circle. Then the measure of

∠ A QB

is

$Circle with tangents PA and PB from external point P$

Q22.Assertion-Reason1 mark(2024)Medium

**Assertion (A):** The tangents drawn at the end points of a diameter of a circle, are parallel.

**Reason (R):** Diameter of a circle is the longest chord.

Q23.MCQ1 mark(2024)Medium

A B

and

C D

are two chords of a circle intersecting at

P

. Choose the correct statement from the following:

$Circle with chords AB and CD intersecting at P$

Q24.MCQ1 mark(2024)Easy

In the given figure, tangents

P A

and

PB

to the circle centred at

O

, from point

P

are perpendicular to each other. If

P A = 5

cm, then length of

A B

is equal to

$Circle with tangents PA and PB perpendicular to each other$

Q25.MCQ1 mark(2024)Easy

In the given figure,

A T

is tangent to a circle centred at

O

. If

∠ C A T = 40°

, then

∠ CB A

is equal to

$Circle with tangent AT and angle CAT = 40 degrees$

Q26.MCQ1 mark(2024)Easy

In the given figure,

A B

and

A C

are tangents to the circle. If

∠ A BC = 42°

, then the measure of

∠ B A C

is:

$Circle with tangents AB and AC, angle ABC = 42 degrees$

Q27.MCQ1 mark(2024)Easy

In the given figure,

QR

is a common tangent to the two given circles touching externally at

A

. The tangent at

A

meets

QR

P

. If

A P = 4.2

cm, then the length of

QR

is:

$Two circles touching externally at A with common tangent QR$

Q28.MCQ1 mark(2024)Easy

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

$Circle with chord AB subtending right angle at centre O$

Q29.MCQ1 mark(2024)Easy

In the given figure,

K J

and

K L

are two tangents to the circle. If

∠ K J L = 42°

, then the measure of

∠ J O L

is:

Q30.MCQ1 mark(2025)Easy

The tangents drawn at the extremities of the diameter of a circle are always:

Q31.MCQ1 mark(2025)Easy

In the given figure, PA is a tangent from an external point P to a circle with centre O. If

∠ POB = 115°

, then

∠ A PO

is equal to:

$Circle with tangent PA and centre O$

Q32.MCQ1 mark(2025)Medium

In the given figure, RS is the tangent to the circle at the point L and MN is the diameter. If

∠ NM L = 30°

, then

∠ R L M

is:

$Circle with diameter MN, tangent RS at point L, angle NML = 30°$

Q33.MCQ1 mark(2025)Easy

If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other at an angle of

80°

as shown in the given figure, then the measure of

∠ PO A

is:

$Circle with tangents PA and PB from external point P, angle APB = 80°$

Q34.Assertion-Reason1 mark(2025)Easy

Assertion (A): If two tangents are drawn to a circle from an external point, then they subtend equal angles at the centre of the circle.

Reason (R): A parallelogram circumscribing a circle is a rhombus.

Q35.MCQ1 mark(2025)Medium

In the adjoining figure, AB is the chord of the larger circle touching the smaller circle. The centre of both the circles is O. If

A B = 2 r

and

OP = r

, then the radius of larger circle is:

$Concentric circles with centre O, chord AB tangent to inner circle at P$

Q36.MCQ1 mark(2025)Easy

A parallelogram having one of its sides 5 cm circumscribes a circle. The perimeter of parallelogram is:

Q37.Assertion-Reason1 mark(2025)Medium

Assertion (A): Tangents drawn at the end points of a diameter of a circle are always parallel to each other.
Reason (R): The lengths of tangents drawn to a circle from a point outside the circle are always equal.

Q38.VSA2 marks(2020)Medium

In the given figure, two tangents

TP

and

TQ

are drawn to a circle with centre

O

from an external point

T

. Prove that

∠ PTQ = 2∠ OPQ

.

$Circle with centre O, tangents TP and TQ from external point T$

Q39.VSA2 marks(2020)Medium

In Fig. 4,

A B

is a chord of circle with centre

O

A OC

is diameter and

A T

is tangent at

A

. Prove that

∠ B A T = ∠ A CB

.

$Circle with chord AB, diameter AOC, tangent AT$

Q40.VSA2 marks(2020)Medium

In Figure 6, a quadrilateral

A BC D

is drawn to circumscribe a circle. Prove that

A B + C D = BC + A D

.

$Quadrilateral ABCD circumscribing a circle$

Q41.VSA2 marks(2020)Medium

In Figure 7, find the perimeter of

△ A BC

, if

A P = 12

cm.

$Triangle ABC with incircle, tangent points B, C on sides, P and Q on extensions$

Q42.VSA2 marks(2022)Easy

Two concentric circles are of radii

4

cm and

3

cm. Find the length of the chord of the larger circle which touches the smaller circle.

Q43.VSA2 marks(2022)Medium

In Fig. 1,

A B

is diameter of a circle centered at

O

BC

is tangent to the circle at

B

. If

OP

bisects the chord

A D

and

∠ A OP = 60°

, then find

m ∠ C

.

$Circle with diameter AB, tangent BC at B, chord AD bisected by OP$

Q44.VSA2 marks(2022)Medium

In Fig. 2,

X A Y

is a tangent to the circle centered at

O

. If

∠ A BO = 40°

, then find

m ∠ B A Y

and

m ∠ A OB

.

$Circle with tangent XAY at point A, center O, point B on circle$

Q45.VSA2 marks(2022)Medium

In Fig. 1, there are two concentric circles with centre O. If

A RC

and

A QB

are tangents to the smaller circle from the point

A

lying on the larger circle, find the length of

A C

, if

A Q = 5

cm.

$Two concentric circles with centre O, tangents ARC and AQB from point A$

Q46.Case-Study2 marks(2023)Medium

OR part of Q38(c): Find the length of

PQ

Q47.VSA2 marks(2023)Easy

In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If

∠ B A C = 65°

, then find the measure of

∠ BOC

Q48.VSA2 marks(2023)Medium

In the given figure,

PT

is a tangent to the circle centered at

O

OC

is perpendicular to chord

A B

. Prove that

P A \cdot PB = P C^{2} - A C^{2}

.

$Circle with tangent PT and chord AB$

Q49.VSA2 marks(2023)Medium

In the given figure,

P A

is a tangent to the circle drawn from the external point

P

and

PBC

is the secant to the circle with

BC

as diameter. If

∠ A OC = 130°

, then find the measure of

∠ A PB

, where

O

is the centre of the circle.

Q50.VSA2 marks(2024)Medium

In the given figure,

A B

and

C D

are tangents to a circle centred at

O

. Is

∠ B A C = ∠ D C A

? Justify your answer.

$Circle with tangents AB and CD from points A and C$

Q51.VSA2 marks(2024)Medium

In the given figure,

O

is the centre of the circle. If

∠ A OB = 145°

, then find the value of

x

.

$Circle with centre O, angle AOB = 145 degrees, angle x at C$

Q52.VSA2 marks(2024)Easy

If two tangents inclined at an angle of

60°

are drawn to a circle of radius 3 cm, then find the length of each tangent.

Q53.VSA2 marks(2024)Easy

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

Q54.VSA2 marks(2025)Easy

A person is standing at P outside a circular ground at a distance of

26

m from the centre of the ground. He found that his distances from the points A and B on the ground are

10

m (PA and PB are tangents to the circle). Find the radius of the circular ground.

$Circle with tangents from external point P$

Q55.VSA2 marks(2025)Medium

At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY at a distance of 16 cm from A.

Q56.VSA2 marks(2025)Easy

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

Q57.SA3 marks(2020)Medium

If a circle touches the side

BC

of a triangle

A BC

P

and extended sides

A B

and

A C

Q

and

R

, respectively, prove that

A Q = \frac{1}{2} (BC + C A + A B)

.

$Circle touching triangle and extended sides$

Q58.SA3 marks(2020)Medium

Prove that, a tangent to a circle is perpendicular to the radius through the point of contact.

Q59.SA3 marks(2020)Medium

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 segment joining the points of contact at the centre.

$Circle with tangents from external point P$

Q60.SA3 marks(2020)Hard

In Figure-7,

X Y

and

MN

are two parallel tangents to a circle with centre

O

and another tangent

A B

with point of contact

C

intersecting

X Y

A

and

MN

B

. Prove that

∠ A OB = 90°

.

$Circle with parallel tangents XY and MN$

Q61.SA3 marks(2020)Medium

Draw a pair of tangents to a circle of radius 3 cm which are inclined to each other at an angle of

60°

Q62.SA3 marks(2022)Medium

Draw a circle of radius

3

cm. Take two points

P

and

Q

on one of its extended diameter each at a distance of

7

cm from its centre. Construct tangents to the circle from these two points

P

and

Q

Q63.SA3 marks(2022)Medium

Draw two concentric circles of radii

2

cm and

5

cm. From a point on the outer circle, construct a pair of tangents to the inner circle.

Q64.SA3 marks(2022)Medium

Construct a pair of tangents to a circle of radius 4 cm which are inclined to each other at an angle of

60°

Q65.SA3 marks(2023)Medium

Two tangents

TP

and

TQ

are drawn to a circle with centre

O

from an external point

T

. Prove that

∠ PTQ = 2∠ OPQ

.

$Circle with centre O, tangents TP and TQ from external point T$

Q66.SA3 marks(2023)Medium

In the given figure, a circle is inscribed in a quadrilateral

A BC D

in which

∠ B = 90°

. If

A D = 17

cm,

A B = 20

cm and

D S = 3

cm, then find the radius of the circle.

$Quadrilateral ABCD with inscribed circle, angle B = 90 degrees, points D, S, C, P, B, Q, A, R marked$

Q67.SA3 marks(2023)Medium

In the given figure, O is the centre of the circle and QPR is a tangent to it at P. Prove that

∠ Q A P + ∠ A PR = 90°

Q68.SA3 marks(2023)Medium

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-segment joining the points of contact at the centre.

Q69.SA3 marks(2023)Medium

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

Q70.SA3 marks(2024)Hard

In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle. Prove that

∠ POQ = 90°

.

$Circle with diameter AB and tangents AQ, BP, PQ$

Q71.SA3 marks(2024)Medium

A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If

A D

is perpendicular to

D C

BC = 30

cm and

BS = 24

cm, then find the length

D C

.

$Circle inscribed in quadrilateral ABCD$

Q72.SA3 marks(2024)Hard

In the given figure,

PQ

is tangent to a circle centred at

O

and

∠ B A Q = 30°

; show that

BP = BQ

.

$Circle with tangent PQ and angle BAQ = 30 degrees$

Q73.SA3 marks(2024)Hard

In the given figure,

A B

BC

C D

and

D A

are tangents to the circle with centre

O

forming a quadrilateral

A BC D

.

Show that

∠ A OB + ∠ CO D = 180°

.

$Quadrilateral ABCD with inscribed circle centred at O$

Q74.SA3 marks(2024)Medium

Prove that the parallelogram circumscribing a circle is a rhombus.

Q75.SA3 marks(2024)Medium

Prove that the parallelogram circumscribing a circle is a rhombus.

Q76.SA3 marks(2025)Medium

In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that

∠ B A C + ∠ A C D = 90°

Q77.SA3 marks(2025)Hard

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

Q78.SA3 marks(2025)Medium

Prove that the parallelogram circumscribing a circle is a rhombus.

$Parallelogram ABCD circumscribing a circle with tangent points P, Q, R, S$

Q79.SA3 marks(2025)Medium

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-segment joining the points of contact at the centre.

$Tangents PA and PB from external point P to circle with centre O$

Q80.SA3 marks(2025)Hard

In the given figure, PC is a tangent to the circle at C. AOB is the diameter which when extended meets the tangent at P. Find

∠ CB A

and

∠ BCO

, if

∠ PC A = 110°

.

$Circle with tangent PC at C, diameter AOB extended to P$

Q81.SA3 marks(2025)Medium

Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimeter of ABCD.

Q82.LA4 marks(2020)Medium

Draw a line segment

A B

of length

7

cm. Taking

A

as centre, draw a circle of radius

3

cm and taking

B

as centre, draw another circle of radius

2

cm. Construct tangents to each circle from the centre of the other circle.

Q83.LA4 marks(2020)Medium

Draw a circle of radius 2 cm with centre

O

and take a point

P

outside the circle such that

OP = 6.5

cm. From

P

, draw two tangents to the circle.

Q84.LA4 marks(2022)Hard

In Figure 1, a triangle

A BC

with

∠ B = 90°

is shown. Taking

A B

as diameter, a circle has been drawn intersecting

A C

at point

P

. Prove that the tangent drawn at point

P

bisects

BC

.

$Triangle ABC with angle B = 90 degrees, circle on AB as diameter intersecting AC at P$

Q85.LA4 marks(2022)Hard

In Fig. 4,

PQ

is a chord of length

8

cm of a circle of radius

5

cm. The tangents at

P

and

Q

meet at a point

T

. Find the length of

TP

.

$Circle with chord PQ, tangents at P and Q meeting at point T$

Q86.LA4 marks(2022)Medium

In Fig. 2, if a circle touches the side

QR

△ PQR

S

and extended sides

PQ

and

PR

M

and

N

, respectively, then prove that

PM = \frac{1}{2} (PQ + QR + PR)

.

$Circle touching side QR of triangle PQR at S, extended sides PQ and PR at M and N$

Q87.LA4 marks(2022)Hard

In Fig. 3, a triangle

A BC

is drawn to circumscribe a circle of radius 4 cm such that the segments

B D

and

D C

into which

BC

is divided by the point of contact

D

are of lengths 6 cm and 8 cm respectively. If the area of

△ A BC

84

^{2}

, find the lengths of sides

A B

and

A C

.

$Triangle ABC with inscribed circle, point D on BC with BD = 6 cm and DC = 8 cm$

Q88.Case-Study4 marks(2023)Medium

The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around one and a half times through a circle, then releases the throw. When released, the discus travels along tangent to the circular spin orbit.

$Discus thrower with tangent line$

In the given figure,

A B

is one such tangent to a circle of radius 75 cm. Point

O

is centre of the circle and

∠ A BO = 30°

PQ

is parallel to

O A

.

$Circle with centre O, radius 75 cm, tangent AB, angle ABO = 30 degrees, PQ parallel to OA$

Based on above information:
(a) Find the length of

A B

.
(b) Find the length of

OB

.
(c) Find the length of

A P

.

OR

(c) Find the length of

PQ

Q89.Case-Study4 marks(2024)Hard

A backyard is in the shape of a triangle

A BC

with right angle at

B

A B = 7

m and

BC = 15

m. A circular pit was dug inside it such that it touches the walls

A C

BC

and

A B

P

Q

and

R

respectively such that

A P = x

m.

$Triangle ABC with inscribed circle$

Based on the above information, answer the following questions:
(i) Find the length of

A R

in terms of

x

.
(ii) Write the type of quadrilateral

BQOR

.
(iii) (a) Find the length

PC

in terms of

x

and hence find the value of

x

.

OR

(b) Find

x

and hence find the radius

r

of the circle.

Q90.LA5 marks(2023)Hard

A triangle

A BC

is drawn to circumscribe a circle of radius

4

cm such that the segments

B D

and

D C

are of lengths

10

cm and

8

cm respectively. Find the lengths of the sides

A B

and

A C

, if it is given that area

△ A BC = 90

^{2}

.

$Triangle ABC circumscribing a circle$

Q91.LA5 marks(2023)Hard

Two circles with centres

O

and

O^{'}

of radii

6

cm and

8

cm, respectively intersect at two points

P

and

Q

such that

OP

and

O^{'} P

are tangents to the two circles. Find the length of the common chord

PQ

Q92.LA5 marks(2023)Medium

Two tangents

TP

and

TQ

are drawn to a circle with centre

O

from an external point

T

. Prove that

∠ PTQ = 2∠ OPQ

.

$Circle with tangents TP and TQ from external point T$

Q93.LA5 marks(2023)Medium

A circle touches the side

BC

of a

△ A BC

at a point

P

and touches

A B

and

A C

when produced at

Q

and

R

respectively. Show that

A Q = \frac{1}{2}

(Perimeter of

△ A BC

).

$Triangle ABC with circle touching BC at P and sides produced at Q and R$

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