# NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.6 – Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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

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

- PQ and ₹ are diameters
- PRQS is a rectangle.

Solution:

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

Given,

- PQ and ₹ 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,