# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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) ,