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 Equation and Equation . If S is a point on the circle other than the arc PQR, find Equation .

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, Equation and Equation

Now,

  • Equation

  • Equation ( Equation and Equation )

  • Equation

  • 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, Equation

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 Equation , Equation Radius of the circle.

  • Thus, Equation is an equilateral triangle, and, Equation

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° Equation

Since, PQRD is a cyclic quadrilateral,

  • Equation (Opposite angles of cyclic quadrilateral)

  • Equation ( Equation )

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

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