NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.5 – Part 4

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Vertically opposite anges formed by two intersecting lines

Vertically Opposite Angles

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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

Two Circles Intersecting at Two Points Q and B


  • 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.


Two Circles with Diameters on Triangle

  • 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.