NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.5 – Part 4
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Vertical opposite angles are the angles that are vertically opposite to each other when two lines intersect
Q9 Two circles intersect at two points Q and B. Through Q, two line segments PQA and RQS are drawn to intersect the circles at P, A and R, and S respectively (see Fig.). Prove that
Solution:

Here, two circles intersect at two point Q and B.

Two line segment PQA and RQS are drawn to intersect the circles at A,D and P,Q respectively

Now, join PR and AS chords.
For chord PR,
(Angles in the same segment) ………equation (1)
For chord AS,

(Angles in same segment) …………equation (2)

PQA and RQS are line segments intersecting at Q, therefore (Vertically opposite angles)………….equation (3)
By the equations (1)


( From equation (3))

( From equation (2))
Q10 If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
Solution:

Given, two circles are drawn on the sides PQ and PR of the triangle as diameters.

The circles intersect at P and S.
To prove, S is on QR. We have to prove that QSR is a straight line, that is, are supplementary.
(Angle in the semi circle)
Therefore, and is straight line.
Thus, S on the QR.