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

Angle ABC bisected by line segment BE

BE Is Angle Bisector of Angle ABC

Angle ABC bisected by line segment BE

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:

Circle c Circle c: Circle through B with center O Segment f Segment f: Segment [P, R] Segment g Segment g: Segment [R, Q] Segment h Segment h: Segment [Q, S] Segment i Segment i: Segment [P, S] Segment j Segment j: Segment [P, Q] Segment k Segment k: Segment [R, S] Point O O = (0.44, 2.76) Point O O = (0.44, 2.76) Point O O = (0.44, 2.76) Point P Point P: Point on c Point P Point P: Point on c Point P Point P: Point on c Point R Point R: Point on c Point R Point R: Point on c Point R Point R: Point on c Point Q Point Q: Point on c Point Q Point Q: Point on c Point Q Point Q: Point on c Point S Point S: Point on c Point S Point S: Point on c Point S Point S: Point on c

PQ and RS Bisect Each Other at Point O

PQ and RS are chords of circle which bisect each other at O.

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 Equation

  • Equation (O is the mid-point of SR)

  • Equation (Vertically opposite angles)

  • Equation (O is the mid-point of SQ)

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

  • Therefore, Equation

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

Similarly, in Equation and Equation

  • Equation ……….equation (2)

From equation (1) and (2)

  • Equation

  • Equation

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.

Equation (Proved above), therefore, Equation

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

Similarly, Equation and Equation

Since Equation and Equation , Therefore PRQS is a cyclic parallelogram

Also, Equation ……….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 ……….equation (4)

From equation (3) and (4)

  • Equation ( Equation )

  • Equation

  • Equation

Since opposite angles of parallelogram PRQS are Equation 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 Equation

Solution:

Circle c Circle c: Circle through R with center O Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [Q, A] Segment h Segment h: Segment [P, C] Segment i Segment i: Segment [P, R] Segment j Segment j: Segment [Q, R] Segment k Segment k: Segment [A, B] Segment l Segment l: Segment [B, C] Segment m Segment m: Segment [C, A] Segment n Segment n: Segment [B, R] Point O O = (0.8, 2.7) Point O O = (0.8, 2.7) Point O O = (0.8, 2.7) Point R R = (0.92, 0.64) Point R R = (0.92, 0.64) Point R R = (0.92, 0.64) Point P Point P: Point on c Point P Point P: Point on c Point P Point P: Point on c Point Q Point Q: Point on c Point Q Point Q: Point on c Point Q Point Q: Point on c Point A Point A: Point on c Point A Point A: Point on c Point A Point A: Point on c Point C Point C: Point on c Point C Point C: Point on c Point C Point C: Point on c Point B Point B: Point on c Point B Point B: Point on c Point B Point B: Point on c

Bisectors of Triangle PQR Intersect at Its Circumcircle

Bisectors of a triangle PQR intersect at its circumcircle centered at O

  • Equation

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

  • Therefore, Equation

Similarly,

  • Equation and Equation

Also,

  • Equation ( Equation )

  • Equation ( Equation )

  • Equation ( Equation )

Now,

  • Equation

  • Equation

  • Equation

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