#### Chapter 16 CirclesÂ R.D. Sharma Solutions for Class 9th Exercise 16.2

**Exercise 16.2**

1.Â The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.

**Solution**

2. Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.

**Solution**

2.Â Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.

**Solution**

3. Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.

**Solution**

4. Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel, If the distance between AB and CD is 3 cm, find the radius of the circle.

**Solution**

5. Give a method to find the centre of a given circle.

**Solution**

Â LetÂ

*A*,Â*B*Â andÂ*C*Â are three distinct points on a circleÂ .
Now joinÂ

*AB*Â andÂ*BC*Â and draw their perpendicular bisectors.
The point of intersection of the perpendicular bisectors is the centre of given circle.

HenceÂ

*O*Â is the centre of circle C(0,r).
6. Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

**Solution**

7. Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

Â

**Solution**
8. Prove that two different circles cannot intersect each other at more than two points .

**Solution**

We have to prove that two different circles cannot intersect each other at more than two points.

Let the two circles intersect in three points A, B and C.

Then as we know that these three points A, B and C are non-collinear. So, a unique circle passes through these three points.

This is a contradiction to the fact that two given circles are passing through A, B, C.

Hence, two circles cannot intersect each other at more than two points.

Hence, proved.

9. A line segment AB is of length 5cm. 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 reason in support of your answer.

**Solution**

Given that, a line AB = 5 cm, one circle having radius r1 = 4 cm which is passing through point A and Band other circle of radius r2 = 2 cm.Â

As we know that the largest chord of any circle is equal to the diameter of that circle.Â

So, 2Ã—r2<ABÂ

There is no possibility to draw a circle whose diameter is smaller than the length of the chord.Â

10. An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.

**Solution**

11. Given an arc of a circle, complete the circle.

**Solution**

Let PQ be an arc of the circle.

In order to complete the circle. First of all we have to find out its centre and radius.

Now take a point R on the arc PQ and join PR and QR.

Draw the perpendicular bisectors of PR and QR respectively.

Let these perpendicular bisectors intersect at point O.

Then OP = OQ, draw a circle with centre O and radius OP = OQ to get the required circle

12. Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

**Solution**

13. Suppose you are given a circle. Give a construction to find its centre.

**Solution**

14. Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are opposite side of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

**Solution**

15. The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?

**Solution**