# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Q-9 Two congruent circles intersect each other at point P and Q. Through P any line segment APB is drawn so that A, B lie on the two circles. Prove that QA = QB.

Solution:

Given,

- Two congruent circles intersecting each other at point P and Q.
- PQ is a common chord of these circles.

Since PQ is common chord

- Angle subtended by arcs of equal lengths on congruence circles must be equal, therefore,
- Therefore in isosceles triangle AQB,

Q-10 In any triangle PQR, if the angle bisector of and perpendicular bisector of QR intersect, prove that they intersect on the circumcircle of the circumcircle of triangle PQR.

Solution:

Given,

- PQR is a triangle inscribed in a circle with centre at O, A is a point on the circle
- PA is the internal bisector of and B is the mid-point of QR

Prove:

BA is the right bisector of QR

In , since both the arcs QA and AR subtend equal angle at the circumference (.

Therefore in triangle QAR,

- (Given)
- (common line)

Therefore, by SSS criterion of congruence,

Hence, (corresponding parts of congruent triangles)

Now,

- ()

Therefore, BA is the right bisector of QR.