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# NCERT solutions Mathematics Class 10 chapter 10 Circles

## Chapter 10 - Circles

#### Page 209

How many tangents can a circle have?

Q 1 | Page 209

Fill in the blank:

A tangent to a circle intersects it in _______ point (s).

Q 2.1 | Page 209

Fill in the blank:

A line intersecting a circle in two points is called a __________.

Q 2.2 | Page 209

Fill in the blank:

A circle can have __________ parallel tangents at the most.

Q 2.3 | Page 209

Fill in the blank:

The common point of a tangent to a circle and the circle is called ____

Q 2.4 | Page 209

A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :

(A) 12 cm.

(B) 13 cm

(C) 8.5 cm

(D) sqrt119 cm test

Q 3 | Page 209

Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Q 4 | Page 209

#### Pages 213 - 214

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is
(A) 7 cm

(B) 12 cm

(C) 15 cm

(D) 24.5 cm

Q 1 | Page 213

In the given figure, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to

(A) 60°

(B) 70°

(C) 80°

(D) 90°

Q 2 | Page 213

If tangents PA and PB from a point P to a circle with centre O are inclined to each other an angle of 80°, then ∠POA is equal to

(A) 50°

(B) 60°

(C) 70°

(D) 80°

Q 3 | Page 213

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

Q 4 | Page 214

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

Q 5 | Page 214

The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Q 6 | Page 214

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

Q 7 | Page 214

In Fig.2, a quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB + CD = BC + DA.

Q 8 | Page 214

In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB=90°

Q 9 | Page 214

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 pointsof contact to the centre.

Q 10 | Page 214

Prove that a parallelogram circumscribing a circle is a rhombus.

Q 11 | Page 214

Prove that the paralleogram circumscribing a circle, is a rhombus

Q 11 | Page 214

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see given figure). Find the sides AB and AC.

Q 12 | Page 214

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

Q 13 | Page 214

#### Extra questions

In Fig., if AB = AC, prove that BE = EC

A point P is 13 cm from the centre of the circle. The length of the tangent drawn from P to the circle is 12cm. Find the radius of the circle.

Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25 cm. Given that the radius of the circle is 7 cm.

Prove that the tangents at the extremities of any chord make equal angles with the chord.

PA and PB are tangents from P to the circle with centre O. At point M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.

In fig. XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that, XA + AR = XB + BR.

In the given figure, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects PQ at A and RS at B. Prove that ∠AOB = 90º

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

Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.

Prove that there is one and only one tangent at any point on the circumference of a circle.

ABC is a right triangle, right angled at B. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6 cm and 8 cm. Find the radius of the incircle.

ABCD is a quadrilateral such that ∠D = 90°. A circle (O, r) touches the sides AB, BC, CD and DA at P,Q,R and If BC = 38 cm, CD = 25 cm and BP = 27 cm, find r.

In two concentric circles, prove that all chords of the outer circle which touch the inner circle are of equal length.

In fig., circles C(O, r) and C(O’, r/2) touch internally at a point A and AB is a chord of the circle C (O, r) intersecting C(O’, r/2) at C, Prove that AC = CB.

In fig., O is the centre of the circle, PA and PB are tangent segments. Show that the quadrilateral AOBP is cyclic.

A quadrilateral ABCD is drawn to circumscribe a circle, as shown in the figure. Prove that AB + CD = AD + BC

A circle touches the side BC of a ∆ABC at P, and touches AB and AC produced at Q and R respectively, as shown in the figure. Show that AQ = \frac { 1 }{ 2 }

In the given figure, the incircle of ∆ABC touches the sides BC, CA and AB at D, E, F respectively. Prove that AF + BD + CE = AE + CD + BF = \frac { 1 }{ 2 } ("perimeter of ∆ABC")

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.

In the given figure, PQ is a chord of length 8cm of a circle of radius 5cm. The tangents at P and Q intersect at a point T. Find the length TP

From a point P, 10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn. Find the radius of the circle.

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