NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 1
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Theorems of the circle
Q1 Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.
Solution:
Given
Two circles with centres P and R, which intersect each other at Q and S
Construction
Join PQ, PS, SR, QR.
In and

(radius of the same circle)

(radius of the same circle)

(common line)

Therefore by SideSideSide criterion of congruence,
Therefore, (corresponding parts of congruent triangles)
Q2 Two chords PQ and RS of lengths 5cm and 11cm respectively of a circle are parallel to each and are on opposite sides of its centre. If the distance between PQ and RS is 6cm, find the radius of the circle
Solution:
Given,

O is the centre of the circle

And its radius be r cm

and

Construction,

Join OP and OR

Since both P and R are on the circle and O is the center,
and and , Therefore, B, O, A are collinear.
Let , since . Therefore,
Since, the perpendicular from the centre to a chord of the circle bisects the chord, therefore,

()

()
In right



………….equation (1)
In right




…………equation (2)
From equation (1) and (2)
Now, putting in equation (1)
So, the radius of the circle is .