Home
Class 9
MATHS
/P Q R =100^(@), where P, Q and R are po...

`/_P Q R =100^(@)`, where P, Q and R are points on a circle with centre O. Find `/_O P R`

Text Solution

Verified by Experts

Since all points `P, Q, R, S` lie on the circle, `PQRS` becomes a cyclic quadrilateral.
We know that the sum of either pair of opposite angles of a cyclic quadrilateral is `180^@`
Therefore,`anglePQR + anglePSR = 180^@`
`100^@ + anglePSR = 180^@`
`anglePSR = 180^@ - 100^@ = 80^@`
We know that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle
Therefore,`anglePOR = 2anglePSR`
= `2 × 80^@`
= `160^@`
Consider the `triangleOPR`. It is an isosceles triangle as `OP = OR` = Radius of the circle.
Thus, `angleOPR = angleORP`
Sum of all angles in a triangle is `180^@`
Therefore, `angleOPR + anglePOR + angleORP = 180^@`
`angleOPR + 160^@ + angleOPR = 180^@`
`2angleOPR = 180^@ - 160^@`
`angleOPR = 10^@`
Promotional Banner

Topper's Solved these Questions

  • CIRCLES

    NCERT ENGLISH|Exercise Solved Examples|6 Videos

  • CIRCLES

    NCERT ENGLISH|Exercise EXERCISE 10.2|2 Videos

  • CIRCLES

    NCERT ENGLISH|Exercise Exercise 10.1|2 Videos

  • AREAS OF PARALLELOGRAMS AND TRIANGLES

    NCERT ENGLISH|Exercise Solved Examples|4 Videos

  • CONSTRUCTIONS

    NCERT ENGLISH|Exercise EXERCISE 11.2|5 Videos

Similar Questions

Explore conceptually related problems

In Figure, /_P Q R=100^0, where P ,\ Q\ a n d\ R are points on a circle with centre. Odot Find /_O P R

In Fig. 10.101, P Q and P R are two tangents to a circle with centre O . If /_Q P R=46^o , then /_Q O R equals (a) 67^o (b) 134^o (c) 44^o (d) 46^o

In Fig. P Q is tangent at a point R of the circle with centre O . If /_T R Q=30^o , find /_P R S . (FIGURE)

P Q is a tangent drawn from a point P to a circle with centre O and Q O R is a diameter of the circle such that /_P O R=120o , then /_O P Q is 60o (b) 45o (c) 30o (d) 90o

If TP and TQ are the two tangents to a circle with centre O so that /_P O Q=110^@ , then /_P T Q is equal to

If the tangent at a point P to a circle with centre O cuts a line through O at Q such that P Q=24 c m and O Q=25 c m . Find the radius of the circle.

P Q is a tangent to a circle with centre O at the point P . If /_\ O P Q is an isosceles triangle, then /_O Q P is equal to :

In Figure, A P B is a tangent to a circle with centre O at point P . If /_Q P B=50^@ , then the measure of /_P O Q is (a) 100^@ (b) 120^@ (c) 140^@ (d) 150^@

Two tangents T P and T Q are drawn to a circle with centre O from an external point T . Prove that /_P T Q=2/_O P Q .

Two tangents T P and T Q are drawn to a circle with centre O from an external point T . Prove that /_P T Q=2/_O P Q .