Page 12.25

### EXERCISE 12.1

**Type 1**

#### QUESTION 1

**(i) How many circles can be drawn through three non-collinear points ?**

(a) 1

(b) 2

(c) 3

(d) infinite

**(ii) Of the angles subtended by the equal chords at the center of the same circle ?**

(a) One is equal to the other

(b) One is half of the other

(c) One is four times the other

(d) One is two times the other

**(iii) O and O’ are the centers of two congruent circles . Degree measure of are AB of first circle is 50° and that of arc A’B’ of the other circle is 75° , then what is the ratio of arcs AB and A’B’ ?**

(a) 1:2

(b) 2:3

(c) 3:2

(d) 1:4

**(iv) Two arcs AB , of a circle subtend the same angle of 50° at the centre . What is the relation between the arcs ?**

(a)

(b)

(c)

(d) none of these

**(v) Arcs AB and CD of a circle are such that then what will be the ratio of the lengths of AB and CD ?**

(a) 1:1

(b) 2:3

(c) 1:2

(d) 2:1

**(vi) Two chords AB and CD of a circle are at a distance of 3.5 cm from the centre . Which of the following relation exist between AB and CD ?**

(a) AB>CD

(b) AB=CD

(c) AB<CD

(d) none of these

**(vii) There are four points P, Q, R and S on a circle. Which of the following statements is true ?**

(a) P , Q , S are collinear

(b) P , Q , R are collinear

(c) P , Q , R are non-collinear

(d) Q , R , S are collinear

Sol :

#### QUESTION 2

**Fill in the blanks**

**(i) In a circle with centre O , a chord AB is drawn and M is its mid point which is joined to O , then ∠OMA=__**

**(ii) Two circles are said to be congruent if and only if their __ are equal .**

**(iii) In a circle , if two arcs are congruent then corresponding chords are __**

**(iv) If A ,B ,C be three non-collinear points , then through these three points __ circle can be drawn **

**(v) Chords equidistant from the centre of a circle are __**

**(vi) Of any two chords of a circle the chord nearer to the circle is __ than the other **

**(vii) The perpendicular from the centre of the circle to a chord __ the chord**

**(viii) The longest chord of a circle is __**

#### QUESTION 3

**(i) How many circles can be drawn passing through two points ?**

**(ii) A point C is on the line AB. Can a circle through A ,B ,C be drawn ?**

**(iii) What is that circle which passes through three vertices of any triangle ?**

**(iv) How many circles can be drawn through three vertices A , B and C of a triangle ?**

**(v) If in a circle arc AB = 6 cm, arc CD= 6 cm and ∠AOB=65° , then find ∠COD , where O is the centre of the circle. **

**(vi) If in a circle arc AB=9 cm, arc CD=4.5 cm , ∠AOB=80° , then find ∠COD , where O is the centre of the circle **

**(vii) If in a circle, chord PQ= 7 cm, chord RS=14 cm and ∠POQ=75° , then find the value of ∠ROS , where O is the centre of the circle **

**(viii) What part of the circumference is the length of the. arc Between two radii subtending an angle of 90° at the centre of the circle ?**

Sol :

#### QUESTION 4

**ABC is a triangle, state which of the following statements is true ?**

**(i) A circle can pass through the points A , B and C**

**(ii) Two separate circles can pass through the points A , B and C **

**(iii) Infinite numbers of circles can pass through the points A , B and C**

**(iv) Not a single circle can pass through the points A , B and C**

Sol :

#### QUESTION 5

**(i) Length of the chord of a circle is equal to its radius , then angle subtended by this chord at the centre of the circle is equal to :**

**(a) 90°**

**(b) 30°**

**(c) 60°**

**(d) 120°**

(ii) The minimum distance of 8cm long chord from the centre is 3 cm , then diameter of the circle will be :

(a) 4cm

(b) 3cm

(c) 5cm

(d) 10cm

(iii) The distance of 10cm long chord from the centre of the circle is 12cm then which of the following will be radius of the circle ?

(a) 12cm

(b) 22cm

(c) 13cm

(d) 5cm

Sol :

#### QUESTION 6

**In the adjoining figure, ∠OBA=25° and ∠OCA=35° . Find the ∠BAC**

<fig to be added>

Sol :

#### QUESTION 7

**(i) The radius of any circle is 5cm can perpendicular drawn from the centre to a chord is 4cm , then find the length of the chord.**

**(ii) The length of the chord at a distance of 5 cm from the centre of the centre of the circle is 24 cm , then find the length of that chord which is at a distance of 12 cm from the centre**

Sol :

#### QUESTION 8

**AB and CD are two parallel chords of lengths 5 cm , 11 cm respectively of a circle . If distance between the chords be 3 cm , then find the radius of the circle**

Sol :

#### QUESTION 9

**Length of common chord of two circles in 30 cm . If diameter of a circle is 50 cm and that if another is 34 cm , then find the distance between the centres of the two circles .**

Sol :

#### QUESTION 10

Diameter of a circle is 20 cm . Two parallel chords of lengths 16 cm and 72 cm of this circle are drawn . Find the distance between the two chords when the two chords are :

(i) On the same side of the centre

(ii) On either side of the centre

Sol :

#### QUESTION 11

**Prove that the vertices of equilateral triangle divides the circum-circle in three equal parts.**

[Hint: AB=BD=CA

∴ arc AB= arc BD = arc CA]

Sol :

#### QUESTION 12

Prove that two different circles cannot intersect at more than two points.

[Hint: Let the two circles intersect at three points A, B and C. Then, A, B and C will be non collinear points. But according to theorem 13.5, one and only one circle can be drawn through three non-collinear points A ,B and C . This contradicts our hypothesis

Hence , two separate circles cannot intersect at more than two points]

Sol :

**TYPE 3**

#### QUESTION 13

**If each of the two parallel chords of a circle are bisected by a third chord, then prove that the third chord is the diameter of the circle.**

Sol :

[Hint: In the given figure. let the third chord be PQ. .loin OM and ON

<hint to be added>]

#### QUESTION 14

AB and AC are two equal chords of a circle . Prove that the centre of the circle lies on the bisector of ∠BAC

<hint to be added>

Sol :

#### QUESTION 15

A line segment AB is of length 5 cm. Draw a circle of radius 4 cm passing through A and B . Can you draw a circle of radius 2 cm passing through A and B ? Give reasons in support of your answer.

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 16

In the given figure, AB and BC are two chords of a circle with centre O , such that ∠ABO=∠CBO . Prove that AB=CB

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 17

**AB and CD are two equal chords of a circle with centre O. If M and N are respectively the middle points of chords AB and CD , then prove that ∠AMN=∠CNM**

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 18

**In the figure , Chord PQ is bisected by the diameter AB . If AQ||PB then prove that PQ is also diameter of the circle . **

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 19

**AB and CD are two parallel chords of a circle whose diameter is AC . Prove that AB=CD**

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 20

**AB and CD are two chords of a circle whose center is O and they intersect each other at P . If PO bisects the ∠APD ,then prove that AB=CD**

**OR**

**If two intersecting chords make equal angles with the diameter passing through their point of intersection. Prove that AB=CD.**

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 21

**AB and CD are equal chords of a circle with centre O . If OM⊥AB and ON⊥CD , then prove that ∠OMN=∠ONM . **

[Hint: ]

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 22

**OD is ⊥ on chord AB of the circle with centre O . If BC is a diameter, then prove that CA = 2OD.**

[Hint: ]

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 23

**AB and CD are equal chords of a circle with centre O . Chords on being produced meet at point E. Prove that EB=ED and EA=EC**

[Hint:]

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 24

**Write a method of finding centre of circle**

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 25

**If an arc of a circle be given, then show as to how the circle can be completed .**

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 26

**Of any two chords of a circle, larger chord is nearer to the centre than smaller one .**

<fig to be added>

<hint to be added>

Sol :

#### QUESTION 27

**Of any two chords of a circle , show that one which is nearer to the centre is larger .**

<fig to be added>

<hint to be added>

Sol :