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

PQ and RS are diameters

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 .