#### Chapters

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Co-Ordinate Geometry

Chapter 7: Triangles

Chapter 8: Circles

Chapter 9: Constructions

Chapter 10: Trigonometric Ratios

Chapter 11: Trigonometric Identities

Chapter 12: Trigonometry

Chapter 13: Areas Related to Circles

Chapter 14: Surface Areas and Volumes

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 8: Circles

#### RD Sharma solutions for Class 10 Mathematics Chapter 8 Circles [Pages 34 - 37]

In Fig. 7, two equal circles, with centres O and O’, touch each other at X. OO’ produced meets the circle with centre O’ at A. AC is tangent to the circle with centre O, at the point C. O’D is perpendicular to AC. Find the value of `(DO')/(CO')`

In the fig. ABC is right triangle right angled at B such that BC = 6cm and AB = 8cm. Find the radius of its in circle.

If ΔABC is isosceles with AB = AC and C (0, 2) is the in circle of the ΔABC touching BC at L, prove that L, bisects BC.

In fig.. O is the center of the circle and BCD is tangent to it at C. Prove that ∠BAC +

∠ACD = 90°

In the given figure, PO⊥QO. The tangents to the circle at P and Q intersect at a point T. Prove that PQ and OT are right bisector of each other.

In the fig two tangents AB and AC are drawn to a circle O such that ∠BAC = 120°. Prove that OA = 2AB.

In the given figure, BC is a tangent to the circle with centre O. OE bisects AP. Prove that ΔAEO~Δ ABC.

The lengths of three consecutive sides of a quadrilateral circumscribing a circle are 4cm,5cm and 7cm respectively. Determine the length of fourth side.

In fig common tangents PQ and RS to two circles intersect at A. Prove that PQ = RS.

In figure OQ : PQ = 3 : 4 and perimeter of ΔPDQ = 60cm. determine PQ, QR and OP.

#### RD Sharma solutions for Class 10 Mathematics Chapter 8 Circles Exercise 8.1 [Page 5]

Fill in the blank:

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

Fill in the blank:

A circle can have __________ parallel tangents at the most.

Fill in the blank:

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

Fill in the blank:

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

Fill in the blank

The angle between tangent at a point on a circle and the radius through the point is ........

How many tangents can a circle have?

O is the center of a circle of radius 8cm. The tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB

If the tangent at point P to the circle with center O cuts a line through O at Q such that PQ= 24cm and OQ = 25 cm. Find the radius of circle

#### RD Sharma solutions for Class 10 Mathematics Chapter 8 Circles Exercise 8.2 [Pages 33 - 42]

If PT is a tangent at T to a circle whose center is O and OP = 17 cm, OT = 8 cm. Find the length of tangent segment PT.

Find the length of a tangent drawn to a circle with radius 5cm, from a point 13 cm from the center of the circle.

A point P is 26 cm away from O of circle and the length PT of the tangent drawn from P to the circle is 10 cm. Find the radius of the circle.

If from any point on the common chord of two intersecting circles, tangents be drawn to circles, prove that they are equal.

If the quadrilateral sides touch the circle prove that sum of pair of opposite sides is equal to the sum of other pair.

Out of the two concentric circles , the radius of the outer circle is 5 cm and the chord* AC *of length 8 cm is a tangent to the inner circle . Find the radius of the inner circle .

A chord *PQ *of a circle is parallel to the tangent drawn at a point *R* of the circle . Prove that* R *bisects the arc *PRQ*.

Prove that a diameter *AB *of a circle bisects all those chords which are parallel to the tangent at that point *A* .

If AB, AC, PQ are tangents in Fig. and AB = 5cm find the perimeter of ΔAPQ.

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at center.

In Fig below, PQ is tangent at point R of the circle with center O. If ∠TRQ = 30°. Find

∠PRS.

If PA and PB are tangents from an outside point P. such that PA = 10 cm and ∠APB = 60°. Find the length of chord AB.

In a right triangle *ABC* in which \[\angle\]*B*= 90^{0} , a circle is drawn with *AB *as diameter intersecting the hypotenuse* AC* at *P* . Prove that the tangent to the circle at *P* bisects *BC*.

From an external point P, tangents PA and PB are drawn to the circle with centre O. If CD is the tangent to the circle at point E and PA = 14 cm. Find the perimeter of ABCD.

In the given figure, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. Find the radius of its incircle.

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

From a point P, two tangents PA and PB are drawn to a circle with center O. If OP =

diameter of the circle shows that ΔAPB is equilateral.

Two tangent segments PA and PB are drawn to a circle with center O such that ∠APB =120°. Prove that OP = 2AP

If Δ *ABC* is isosceles with *AB* = *AC* and *C* (*O,* *r*) is the incircle of the Δ*ABC* touching *BC* at *L*,prove that *L* bisects *BC*.

*AB* is a diameter and* *AC is a chord of a circle with centre *O *such that \[\angle BAC = {30}^o\] . The tangent at* C* intersects *AB* at a point *D* . Prove that *BC* =* BD*.

In fig. a circle touches all the four sides of quadrilateral ABCD with AB = 6cm, BC = 7cm, CD = 4cm. Find AD.

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

Two circles touch externally at a point P. from a point T on the tangent at P, tangents TQ and TR are drawn to the circles with points of contact Q and E respectively. Prove that TQ = TR.

*A *is a point at a distance 13 cm from the centre* O* of a circle of radius 5 cm . *AP* and *AQ *are the tangents to the circle at *P* and *Q* . If a tangent* BC* is drawn at a point *R* lying on the minor arc *PQ* to intersect *AP* at *B * and *AQ* at *C* , find the perimeter of the \[∆\]*ABC* .

In the fig. a circle is inscribed in a quadrilateral ABCD in which ∠B = 90° if AD = 23cm,

AB = 29cm and DS = 5cm, find the radius of the circle.

In fig. there are two concentric circles with Centre O of radii 5cm and 3cm. From an

external point P, tangents PA and PB are drawn to these circles if AP = 12cm, find the

tangent length of BP.

In the given figure, *AB* is a chord of length 16 cm of a circle of radius 10 cm. The tangents at *A* and* B* intersect at a point *P*. Find the length of *PA*.

In figure PA and PB are tangents from an external point P to the circle with centre O. LN touches the circle at M. Prove that PL + LM = PN + MN

In the given figure, *BDC* is a tangent to the given circle at point *D* such that *BD* = 30 cm and *CD *= 7 cm. The other tangents *BE* and *CF* are drawn respectively from *B* and *C* to the circle and meet when produced at *A* making BAC a right angle triangle. Calculate (i) *AF*

In the given figure, *BDC* is a tangent to the given circle at point *D* such that *BD* = 30 cm and *CD *= 7 cm. The other tangents *BE* and *CF* are drawn respectively from *B* and *C* to the circle and meet when produced at *A* making BAC a right angle triangle. Calculate (ii) radius of the circle.

If \[d_1 , d_2 ( d_2 > d_1 )\] be the diameters of two concentric circle s and *c* be the length of a chord of a circle which is tangent to the other circle , prove that\[{d_2}^2 = c^2 + {d_1}^2\].

In the given figure, tangents PQ and PR are drawn from an external point to a circle with centre O, such that\[\angle RPQ = 30 ^o\]. A chord RS is drawn parallel to the tangent PQ. Find\[\angle RQS\]

From an external point *P* , tangents *PA *= *PB *are drawn to a circle with centre *O* . If \[\angle PAB = {50}^o\] , then find \[\angle AOB\]

In the given figure, two tangents *AB* and *AC* are drawn to a circle with centre *O* such that ∠*BAC* = 120°. Prove that *OA* = 2*AB*.

The length of three concesutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm, and 7 cm respectively. Determine the length of the fourth side.

The common tangents *AB* and *CD* to two circles with centres *O* and *O'* intersect at *E *between their centres . Prove that the points *O *, *E *and *O'* are collinear .

In the given figure, common tangents *PQ* and *RS* to two circles intersect at *A*. Prove that *PQ* = *RS.*

Two concentric circles are of diameters 30 cm and 18 cm. Find the length of the chord of the larger circle which touches the smaller circle.

*AB* and *CD* are common tangents to two circles of equal radii. Prove that *AB* = *CD*.

A triangle PQR is drawn to circumscribe a circle of radius 8 cm such that the segments QT and TR, into which QR is divided by the point of contact T, are of lengths 14 cm and 16 cm respectively. If area of ∆PQR is 336 cm^{2}, find the sides PQ and PR.

In Fig . 10.69, the tangent at a point *C *of a circle and a diameter *AB* when extended intersect at *P* . If \[\angle\]*PCA* =110^{0}, find \[\angle\] *CBA*

*AB* is a chord of a circle with centre *O* , *AOC* is a diameter and *AT* is the tangent at* A* as shown in Fig . 10.70. Prove that \[\angle\]*BAT* = \[\angle\] *ACB*.

In the given figure, a ∆ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 8 cm and 6 cm respectively. Find the lengths of sides AB and AC, when area of ∆ABC is 84 cm^{2}.

In the given figure, AB is a diameter of a circle with centre O and AT is a tangent. If \[\angle\] AOQ = 58º, find \[\angle\] ATQ.

In the given figure, OQ : PQ = 3.4 and perimeter of Δ POQ = 60 cm. Determine PQ, QR and OP.

Equal circles with centres O and O' touch each other at X. OO' produced to meet a circle with centre O', at A. AC is a tangent to the circle whose centre is O. O'D is perpendicular to AC. Find the value of\[\frac{DO'}{CO}\]

In the given figure, *BC* is a tangent to the circle with centre *O*. *OE* bisects *AP.* Prove that ΔAEO ∼ Δ ABC.

In the given figure, *PO *\[\perp\] *QO*. The tangents to the circle at *P* and *Q* intersect at a point *T*. Prove that *PQ *and *OT*are right bisector of each other.

In the given figure, O is the centre of the circle and *BCD *is tangent to it at *C*. Prove that ∠*BAC* + ∠*ACD* = 90°.

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines .

In Fig. 8.78, there are two concentric circles with centre O. PRT and PQS are tangents to the inner circle from a point P lying on the outer circle. If PR = 5 cm, find the length of PS.

In Fig. 8.79, PQ is a tangent from an external point P to a circle with centre O and OP cuts the circle at T and QOR is a diameter. If ∠POR = 130° and S is a point on the circle, find ∠1 + ∠2.

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.

## Chapter 8: Circles

## RD Sharma solutions for Class 10 Mathematics chapter 8 - Circles

RD Sharma solutions for Class 10 Mathematics chapter 8 (Circles) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 8 Circles are Theorem - Converse of Tangent at Any Point to the Circle is Perpendicular to the Radius, Number of Tangents from a Point to a Circle, Inscribed Angle Theorem, Intercepted Arc, Inscribed Angle, Property of Sum of Measures of Arcs, Tangent Segment Theorem, Converse of Tangent Theorem, Circles passing through one, two, three points, Theorem of External Division of Chords, Theorem of Internal Division of Chords, Converse of Theorem of the Angle Between Tangent and Secant, Theorem of Angle Between Tangent and Secant, Converse of Cyclic Quadrilateral Theorem, Corollary of Cyclic Quadrilateral Theorem, Theorem of Cyclic Quadrilateral, Corollaries of Inscribed Angle Theorem, Angle Subtended by the Arc to the Point on the Circle, Angle Subtended by the Arc to the Centre, Introduction to an Arc, Touching Circles, Number of Tangents from a Point on a Circle, Tangent to a Circle, Tangents and Its Properties, Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers, Cyclic Properties, Tangent - Secant Theorem, Cyclic Quadrilateral, Introduction to Circles, Circles Examples and Solutions, Introduction to Circles, Tangent to a Circle, Number of Tangents from a Point on a Circle.

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