# 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

Q-9 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))

Q-10 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.