NCERT Class 9 Solutions: Circles (Chapter 10) Exercise 10.5 – Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)

Get unlimited access to the best preparation resource for NSO-Level-2 Class-9: fully solved questions with step-by-step explanation- practice your way to success.

Opposite Angles of Cyclic Quadiletral Are Supplimentary

Q-1 In the figure, P, Q and R are three points on a circle with center O such that and . If S is a point on the circle other than the arc PQR, find .

P, Q, R Are Three Points on a Circle with Center O

Solution:

Given,

  • P, Q, R are three points on a circle
  • Its center is O
  • Also, and

Now,

  • ( and )

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

Therefore,

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 at a point on the major arc.

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

Solution:

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

  • In , Radius of the circle.
  • Thus, is an equilateral triangle, and,

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 = ∠ POR =

Since, PQRD is a cyclic quadrilateral,

  • (Opposite angles of cyclic quadrilateral)
  • ()
  • Thus the angles subtended by the chord with length equal to the radius are on major arc and on minor arc.