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#### Question 1:

Find the length of the tangent drawn to a circle with radius 8 cm from a point 17 cm away from the centre of the circle. #### 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. Find the radius of the circle. #### Question 3:

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. #### Question 4:

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 ΔPCD. #### Question 5:

A circle is inscribed in ΔABC, touching AB, BC and AC at P, Q and R, respectively. If AB = 10 cm, AR = 7 cm and CR = 5 cm, find the length of BC. #### Question 6:

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

Two tangent segments BC and BD are drawn to a circle with centre O, such that ∠CBD = 120°. Prove that OB = 2BC. #### Question 8:

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 circles, respectively. If PA = 10 cm, find the length of PB up to one decimal place. #### Question 9:

In the given figure, ABCD is a quadrilateral in which ∠D = 90°. A circle C(O, r) touches the sides AB, BC, CD and DA at P, Q, R, S, respectively. If BC = 38 cm, CD = 25 cm and BP = 27 cm, find the value of r.  #### Question 1:

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

(a) 8 cm

#### Question 2:

In a circle of radius 7 cm, tangent PT is drawn from a point P, such that PT = 24 cm. If O is the centre of the circle, then OP = ? (a) 30 cm
(b) 28 cm
(c) 25 cm
(d) 18 cm

(c) 25 cm
The tangent at any point of a circle is perpendicular to the radius at the point of contact.

#### Question 3:

A point P is 26 cm away from the centre of a circle and the length of the tangent drawn from P to the circle is 24 cm. The radius of the circle is (a) 8 cm
(b) 10 cm
(c) 12 cm
(d) 14 cm

(b) 10 cm
Given, point P is 26 cm away from the centre of a circle and the length of the tangent drawn from P to the circle is 24 cm. The centre of the circle is O.
The tangent at any point of a circle is perpendicular to the radius at the point of contact.

#### Question 4:

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 (a) 3 cm
(b) 6 cm
(c)
(d)

(d)

#### Question 5:

If PA and PB are two tangents to a circle with centre O, such that ∠AOB = 110°, find ∠APB. (a) 90°
(b) 60°
(c) 70°
(d) 55°

(c) 70°

#### Question 6:

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°

(b) 50°

#### Question 7:

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°

(c) 40°

#### Question 8:

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 ∠PQT = ? (a) 100°
(b) 90°
(c) 80°
(d) 75°

#### Question 9:

In the given figure, PQ is a chord of a circle with centre O and PT is a tangent at P, such that ∠QPT = 60°. Then, ∠PRQ = ? (a) 135°
(b) 150°
(c) 120°
(d) 110°

#### Question 10:

If the angle between two radii of a circle is 130°, the angle between the tangents at the ends of the radii is (a) 65°
(b) 50°
(c) 40°
(d) 70°

(b) 50°

#### Question 11:

PA and PB are tangents to the circle with centre O, such that ∠APB = 50°. Then, ∠OAB = ? (a) 25°
(b) 30°
(c) 40°
(d) 50°

#### Question 12:

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 quadrilateral PQOR is (a) 60 cm2
(b) 32.5 cm2
(c) 65 cm2
(d) 30 cm2

(a) 60 cm2

#### 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°

(b) 50°

#### Question 14:

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)

(c)

#### Question 15:

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°

(b) 30°

#### Question 16:

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°. Find ∠QSR. (a) 40°
(b) 50°
(c) 60°
(d) 70°

#### Question 17:

In the given figure, O is the centre of two concentric circles of radii 3 cm and 5 cm. AB is a chord of the outer circle, which touches the inner circle. The length of AB is (a) 4 cm
(b) 7 cm
(c) 8 cm
(d) $\sqrt{34}\mathrm{cm}$

(c) 8 cm

#### Question 18:

In the given figure, ΔABC is circumscribed, touching the circle at P, Q and R. AP = 4 cm, BP = 6 cm, AC = 12 cm and BC = x cm. Find x. (a) 10 cm
(b) 14 cm
(c) 18 cm
(d) 12 cm

(b) 14 cm

#### Question 19:

In the given figure, quadrilateral ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 5 cm, BC = 7 cm and CS = 3 cm, AB = ? (a) 9 cm
(b) 10 cm
(c) 12 cm
(d) 8 cm

(a) 9 cm

#### Question 20:

In the given figure, quadrilateral ABCD is circumscribed, touching the circle at P, Q, R and S. If AP = 6 cm, BP = 5 cm, CQ = 3 cm and DR = 4 cm, then the perimeter of quadrilateral ABCD is (a) 18 cm
(b) 27 cm
(c) 36 cm
(d) 32 cm

#### Question 21:

In the given figure, quad. ABCD is circumscribed, touching the circle at P, Q, R and S such that ∠DAB = 90°, If CS = 27 cm and CB = 38 cm and radius of the circle is 10 cm, then AB = ? (a) 28 cm
(b) 21 cm
(c) 19 cm
(d) 17 cm #### Question 22:

In the given figure, ΔABC is right-angled at B, such that BC = 6 cm and AB = 8 cm. A circle with centre O has been inscribed in the triangle. OPAB, OQBC 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

(a) 2 cm

#### Question 23:

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, the radius of the circle with centre A is (a) 1.5 cm
(b) 2 cm
(c) 2.5 cm
(d) 3 cm

(b) 2 cm

#### Question 24:

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, then PD = ? (a)
(b)
(c)
(d) (c)

#### Question 25:

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 (a)
(b) 5 cm
(c)
(d) 7.5 cm

(b) 5 cm
The lengths of tangents drawn from a point to a circle are equal.

#### Question 26:

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 a circle, then one and only one tangent can be drawn to the circle at P.
(c) If a point P lies outside a 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.

(d) A circle can have more than two parallel tangents, parallel to a given line.
This statement is false because there can only be two parallel tangents to the given line in a circle.

#### Question 27:

Which of the following statements is not true?
(a) A tangent to a circle intersects the circle exactly at one point.
(b) The point common to a circle and its tangent is called the point of contact.
(c) The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact.
(d) A straight line can meet a circle at one point only.

(d)A straight line can meet a circle at one point only.
This statement is not true because a straight line that is not a tangent but a secant cuts the circle at two points.

#### Question 28:

Which of the following statements is not true?
(a) A line which intersects a circle at two points, is called a secant of the circle.
(b) A line intersecting a circle at one point only is called a tangent to the circle.
(c) The point at which a line touches the circle is called the point of contact.
(d) A tangent to the circle can be drawn from a point inside the circle.

(d) A tangent to the circle can be drawn from a point inside the circle.
This statement is false because tangents are the lines drawn from an external point to the circle that touch the circle at one point.

#### Question 29:

Assertion (A)
At 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.

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

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

#### Question 30:

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

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

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

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

Assertion :-
We know that if two tangents are drawn to a circle from an external point, they subtend equal angles at the centre.

Reason:- Given, a parallelogram ABCD circumscribes a circle with centre O.
$AB=BC=CD=AD\phantom{\rule{0ex}{0ex}}$
We know that the
tangents drawn from an external point to circle are equal .

Hence, ABCD is a rhombus.

#### Question 31:

Assertion (A)
In the given figyre, a quadrilateral ABCD is drawn to circumscribe a given circle, as shown.
Then, AB + BC = AD + DC.
Figure

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

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

(d) Assertion (A) is false and Reason (R)is true.
Assertion (A) is false

#### 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°

#### Question 2:

If the angle between two radii of a circle is 130°, then the angle between the angles at the ends of the radii is
Figure

(a) 65°
(b) 40°
(c) 50°
(d) 90° #### Question 3:

If tangents PA and PB from a point P to a circle with centre O are drawn, so that ∠APB = 80°, then ∠POA = ? (a) 40°
(b) 50°
(c) 80°
(d) 60°

(b) 50°

#### 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

(b) 10 cm
Since the tangents from an external point are equal, we have:

#### Question 5:

Find the length of the tangent drawn to a circle with radius 7 cm from a point 25 cm away from the centre of the circle. The length of the tangent to the circle is 24 cm.

#### 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. #### Question 7:

In the given figure, PA and PB are tangents, such that PA = 9 cm and ∠APB = 60°. Find the length of chord AB. The lengths of tangents drawn from a point to a circle are equal.
So, PAPB.

The length of chord AB is 9 cm.

#### Question 8:

Two tangents BC and BD are drawn to a circle with centre O, such that ∠CBD = 120°. Prove that OB = 2BC. #### Question 9:

Fill in the blanks.
(i) A line intersecting a circle at 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.

(i) A line intersecting a circle at two distinct points is called a secant.
(ii) A circle can have two parallel tangents at the most.
(iii) The common point of a tangent to a circle and the circle is called the point of contact.
(iv) A circle can have infinite tangents.

#### Question 10:

Prove that the length of two tangents drawn from an external point to a circle are equal. Given two tangents AP and AQ are drawn from a point A to a circle with centre O.

#### Question 11:

Prove that the tangents drawn at the ends of the diameter of a circle are parallel. Now, radius of a circle is perpendicular to the tangent at the point of contact.

#### Question 12:

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

$\mathrm{Given},\mathit{AB}=\mathit{AC}$
We know that the tangents from an external point are equal.

#### Question 13:

If two tangents are drawn to a circle from an external point,show that they subtend equal angles at the centre. Given : A circle with centre O and a point A outside it. Also, AP and AQ are the two tangents to the circle.
:

#### Question 14:

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord. #### Question 15:

Prove that the parallelogram circumscribing a circle is a rhombus. Given, a parallelogram ABCD circumscribes a circle with centre O.
$AB=BC=CD=AD\phantom{\rule{0ex}{0ex}}$
We know that the lengths of tangents drawn from an exterior point to a circle
are equal.

Hence, ABCD is a rhombus.

#### Question 16:

Two concentric circles are of radii 5 cm and 3 cm, respectively. Find the length of the chord of the larger circle that touches the smaller circle. Given: Two circles have the same centre O and AB is a chord of the larger circle touching the
smaller circle at C; also, OA=5 cm and OC=3 cm.

The length of the chord of the larger circle is 8 cm.

#### Question 17:

A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC. We know that the tangents drawn from an external point to a circle are equal.

#### Question 18:

Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. Given, a quadrilateral ABCD circumscribes a circle with centre O.

#### Question 19:

Prove that the angles between the two tangents drawn form an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre. Given, PA and PB are the tangents drawn from a point P to a circle with centre O. Also, the line segments OA and OB are drawn.

We know that the tangent to a circle is perpendicular to the radius through the point of contact.

From (i) and (ii), we get:
$\angle APB+\angle AOB={180}^{0}$

#### Question 20:

A circle touches the side BC of ∆ABC at P and touches AB and AC produced at Q and R, respectively, as shown in the given figure. Show that
Figure