# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 4

Angle Bisector. A line that splits an angle into two equal angles. ("Bisect" means to divide into two equal parts.)

Q-7 PQ and RS are chords of a circle which bisect each other. Prove that

1. PQ and RS are diameters

2. PRQS is a rectangle.

Solution:

Solution (i): To prove PQ and RS are diameters

Given,

• PQ and RS are two chords of a circle which intersect at O.

• They bisects each other at O

Construction,

Join PS, SQ, QR and RP.

In

• (O is the mid-point of SR)

• (Vertically opposite angles)

• (O is the mid-point of SQ)

• Therefore, by Side-Angle-Side criterion of congruence.

• Therefore,

• Since segment PQ and SR are equal, corresponding arcs in the circle are also equal, therefore ………..equation (1)

Similarly, in and

• ……….equation (2)

From equation (1) and (2)

Therefore PQ divides the circle into two parts

Therefore PQ is a diameter.

Similarly, we can prove that PR is a diameter.

Solution (ii): To prove PRQS is a rectangle.

(Proved above), therefore,

With two lines PR and SQ intersected by transversal SR the pair of interior opposite angles are equal, therefore,

Similarly, and

Since and , Therefore PRQS is a cyclic parallelogram

Also, ……….equation (3) (because opposite angle of a parallelogram are equal)

Since PRQS is a cyclic quadrilateral

Therefore pair of opposite angles are supplementary, therefore ……….equation (4)

From equation (3) and (4)

• ()

Since opposite angles of parallelogram PRQS are each, therefore, PRQS is a rectangle.

Q-8 Bisectors of angle P, Q and R of a triangle PQR intersect its circumcircle at C, A and B respectively, Prove that the angles of the triangle CAB are

Solution:

• Since are angles in same segment of circle (AP), therefore . Similarly, . Therefore equation above can be written as (Since QA and RB are angle bisectors of and .

• Therefore,

Similarly,

• and

Also,

• ()

• ()

• ()

Now,

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