# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 2

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Q-3 If two equal chords of a circle intersect within the circle; prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Solution:

Given,

AB and PQ are chords intersecting at S.

, CD is the diameter.

To prove,

Now, Construction,

Draw And OS is joined.

In and ,

(Equal chords are equidistant from the centre)

(Common line)

( by construction)

Therefore, (By Right Angle Hypotenuse congruence condition)

Therefore,

by Corresponding Parts of Congruent Triangles

(Same angle)

Q-4 If a line intersects two concentric circles (circles with the same centre) with centre O. A common chord intersects the circles at A, B, C and D, prove that AB = CD (see Fig.).

Solution:

Given,

Two concentric circles, with centre at O and common chord intersecting at A,B,C and D

Contruction, draw a line OM ⊥ AD, intersecting AD at M.

Now, since and is drawn from center O

OM is bisects of AD as .

…………equation (1)

Since O is also the center of inner circle, therefore OM also bisects BC as .

Therefore, …….equation (2)

From (1) and (2),