NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.4 – Part 2
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Q3 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)
Q4 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),