NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.5 – Part 1

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Opposite angles of cyclic quadiletral are supplimentary

Opposite Angles of Cyclic Quadiletral Are Supplimentary

Opposite angles of cyclic quadiletral are supplimentary

Q-1 In the figure, P, Q and R are three points on a circle with centre O such that QOR=30° and POQ=60° . If S is a point on the circle other than the arc PQR, find PSR .

Circle c Circle c: Circle through S with center O Angle ? Angle ?: Angle between Q, O, Q' Angle ? Angle ?: Angle between Q, O, Q' Angle ? Angle ?: Angle between Q, O, Q' Angle ? Angle ?: Angle between R, O, R' Angle ? Angle ?: Angle between R, O, R' Angle ? Angle ?: Angle between R, O, R' Segment f Segment f: Segment [P, O] Segment g Segment g: Segment [O, R] Segment h Segment h: Segment [P, S] Segment i Segment i: Segment [R, S] Segment j Segment j: Segment [O, Q] Point O O = (-0.12, 2.36) Point O O = (-0.12, 2.36) Point O O = (-0.12, 2.36) Point S S = (1.18, 0.36) Point S S = (1.18, 0.36) Point S S = (1.18, 0.36) 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

P, Q, R Are Three Points on a Circle With Centre O

P, Q, R are three points on a circle with centre O, ∠QOR = 30° and ∠POQ = 60,S is a point on the circle.

Solution:

Given,

  • P,Q,R are three points on a circle

  • It centre is O

  • Also, QOR=30° and POQ=60

Now,

  • POR=POQ+QOR

  • POR=60°+30° ( QOR=30° and POQ=60 )

  • POR=90°

  • We know angle subtend by an arc at the centre is double the angle subtended by the same arch at the any point on the remaining part of the circle.

  • Therefore, PSR=12POR=12×90°=45°

Q-2 A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Circle c Circle c: Circle through R with center O Segment f Segment f: Segment [P, R] Segment g Segment g: Segment [P, O] Segment h Segment h: Segment [O, R] Segment i Segment i: Segment [P, Q] Segment j Segment j: Segment [Q, R] Segment k Segment k: Segment [P, D] Segment l Segment l: Segment [D, R] Point O O = (0.4, 3.16) Point O O = (0.4, 3.16) Point O O = (0.4, 3.16) Point R R = (2.62, 1.56) Point R R = (2.62, 1.56) Point R R = (2.62, 1.56) 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 D Point D: Point on c Point D Point D: Point on c Point D Point D: Point on c

P, Q, R, D Are Points on a Circle

P, Q, R, D are points on circle, PR is equal to the radius of circle, also triangle OPR is in the circle.

Solution:

Given, PR is equal to the radius of the circle.

  • In ΔOPR , OP=OR=PR= Radius of the circle.

  • Thus, ΔOPR is an equilateral triangle, and, POR=60°

Since angle subtended by an arc at any point on the remainder of the circle is half the angle subtended by the same arc at the center. Therefore, ∠PQR = ½ ∠POR = ½ × 60° = 30° (POR=60°)

Since, PQRD is a cyclic quadrilateral,

  • PQR+PDR=180° (Opposite angles of cyclic quadrilateral)

  • PDR=180°30°=150° ( PQR=30° )

  • Thus the angles subtended by the chord with length equal to the radius are 150° on major arc and 30° on minor arc.

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