#### Chapters

#### Balbharati SSC Class 10 Mathematics 2

## Chapter 3 : Circle

#### Page 55

In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is a tangent at A. Answer the following questions.

(1) What is the measure of ∠CAB ? Why ?

(2) What is the distance of point C from line AB? Why ?

(3) *d*(A,B) = 6 cm, find *d*(B,C).

(4) What is the measure of ∠ABC ? Why ?

In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then

(1) What is the length of each tangent segment ?

(2) What is the measure of ∠MRO ?

(3) What is the measure of ∠ MRN ?

Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON.

What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.

#### Page 58

Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres.

Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres.

If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each other – (i) externally (ii) internally.

In the given figure, the circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that – (1) seg AP || seg BQ,

(2) ∆APR ~ ∆RQB, and

(3) Find ∠ RQB if ∠ PAR = 35°

In the given figure, the circles with centres A and B touch each other at E. Line *l *is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm.

#### Pages 63 - 64

In the given figure, points G, D, E, F are concyclic points of a circle with centre C.

∠ ECF = 70°, m(arc DGF) = 200° find m(arc DE) and m(arc DEF).

In the given figure, ∆QRS is an equilateral triangle. Prove that,

(1) arc RS ≅ arc QS ≅ arc QR

(2) m(arc QRS) = 240°.

In the given figure, chord AB ≅ chord CD, Prove that, arc AC ≅ arc BD

#### Pages 73 - 74

In the given figure, in a circle with centre O, length of chord AB is equal to the radius of the circle. Find measure of each of the following.

(1) ∠ AOB (2)∠ ACB

(3) arc AB (4) arc ACB.

In the given figure, ▢PQRS is cyclic. side PQ ≅ side RQ. ∠ PSR = 110°, Find–

(1) measure of ∠ PQR

(2) m(arc PQR)

(3) m(arc QR)

(4) measure of ∠ PRQ

▢MRPN is cyclic, ∠ R = (5*x *– 13)°, ∠ N = (4*x *+ 4)°. Find measures of ∠ R and ∠ N.

In the given figure, seg RS is a diameter of the circle with centre O. Point T lies in the exterior of the circle. Prove that ∠ RTS is an acute angle.

Prove that, any rectangle is a cyclic quadrilateral.

In the given figure, altitudes YZ and XT of ∆WXY intersect at P. Prove that,

(1) ▢WZPT is cyclic.

(2) Points X, Z, T, Y are concyclic.

In the given figure, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠ NMS.

In the given figure, chords AC and DE intersect at B. If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC).

#### Page 82

In the given figure, ray PQ touches the circle at point Q. PQ = 12, PR = 8, find PS and RS.

In the given figure, chord MN and chord RS intersect at point D.

(1) If RD = 15, DS = 4,

MD = 8 find DN

(2) If RS = 18, MD = 9,

DN = 8 find DS

In the given figure, O is the centre of the circle and B is a point of contact. seg OE ⊥ seg AD, AB = 12, AC = 8, find

(1) AD (2) DC (3) DE.

In the given figure, if PQ = 6, QR = 10, PS = 8 find TS.

In the given figure, seg EF is a diameter and seg DF is a tangent segment. The radius of the circle is r. Prove that, DE × GE = 4r^{2}

#### Pages 82 - 84

Four alternative answers for the following question is given. Choose the correct alternative.

(1) Two circles of radii 5.5 cm and 3.3 cm respectively touch each other. What is the distance between their centers ?

Four alternative answers for the following question is given. Choose the correct alternative.

Two circles intersect each other such that each circle passes through the centre of the other. If the distance between their centres is 12, what is the radius of each circle ?

Four alternative answers for the following question is given. Choose the correct alternative.

A circle touches all sides of a parallelogram. So the parallelogram must be a, ................... .

Four alternative answers for the following question is given. Choose the correct alternative.

Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.

Four alternative answers for the following question is given. Choose the correct alternative.

If two circles are touching externally, how many common tangents of them can be drawn?

Four alternative answers for the following question is given. Choose the correct alternative.

∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB).

Four alternative answers for the following question is given. Choose the correct alternative.

Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6, EB = 10, CE = 8, find ED.

Four alternative answers for the following question is given. Choose the correct alternative.

In a cyclic ▢ABCD, twice the measure of ∠A is thrice the measure of ∠C. Find the measure of ∠C?

Four alternative answers for the following question is given. Choose the correct alternative.

Points A, B, C are on a circle, such that m(arc AB) = m(arc BC) = 120°. No point, except point B, is common to the arcs. Which is the type of ∆ABC?

Four alternative answers for the following question is given. Choose the correct alternative.

(10) Seg XZ is a diameter of a circle. Point Y lies in its interior. How many of the following statements are true ? (i) It is not possible that ∠XYZ is an acute angle. (ii) ∠XYZ can’t be a right angle. (iii) ∠XYZ is an obtuse angle. (iv) Can’t make a definite statement for measure of ∠XYZ.

Line *l *touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following.

(1) What is *d*(O, P) = ? Why ?

(2) If *d*(O, Q) = 8 cm, where does the point Q lie ?

(3) If d(OQ) = 15 cm, How many locations of point Q are line on line l? At what distance will each of them be from point P?

In the given figure, M is the centre of the circle and seg KL is a tangent segment.

If MK = 12, KL = \[6\sqrt{3}\] then find –

(1) Radius of the circle.

(2) Measures of ∠K and ∠M.

In the given figure, O is the centre of the ci

rcle. Seg AB, seg AC are tangent segments. Radius of the circle is *r *and *l*(AB) = *r *, Prove that, ▢ABOC is a square.

#### Balbharati SSC Class 10 Mathematics 2

#### Textbook solutions for Class 10th Board Exam

## Balbharati solutions for Class 10th Board Exam Geometry chapter 3 - Circle

Balbharati solutions for Class 10th Board Exam Geometry chapter 3 (Circle) 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 Maharashtra State Board Textbook for SSC Class 10 Mathematics 2 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10th Board Exam Geometry chapter 3 Circle are 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, 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, 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, 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, Theorem - Converse of Tangent at Any Point to the Circle is Perpendicular to the Radius, Number of Tangents from a Point to a Circle.

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