\n\n
From the figure, P R is a chord of the circle.
\n\nTake any point S on the major arc of the circle.
\n\nJoin P S and R S . A quadrilateral P Q R S is formed.
\n\nIf we observe the figure, P Q R S is a cyclic quadrilateral. (∵ All the vertices are on the circle)
\n\n∠ P Q R + ∠ P S R = 180 ° (∵ Opposite angles of a cyclic quadrilateral)
\n\n⇒ ∠ P S R = 180 ° - 100 ° = 80 °
\n\nWe know that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
\n\n∴ ∠ P O R = 2 ∠ P S R = 2 80 ° = 160 °
\n\n\n\nIn △ P O R ,
\n\nO P = O R (radii of the same circle)
\n\n∴ ∠ O P R = ∠ O R P (Angles opposite to the equal sides of a triangle) …(i)
\n\n∠ O P R + ∠ O R P + ∠ P O R = 180 ° (Angle sum property of a triangle)
\n\n⇒ 2 ∠ O P R + 160 ° = 180 ° (by using equation (i))
\n\n⇒ 2 ∠ O P R = 180 ° - 160 ° = 20 °
\n\n∴ ∠ O P R = 10 °
\n\nHence, the value of ∠ OPR = 10 ° .
\n \n"},"encodingFormat":"text/html","position":0,"text":"10 ° "},"comment":{"@type":"Comment","text":"Use the properties that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle and sum of all the interior angles in a triangle is 180 ° .
\n"},"encodingFormat":"text/markdown","learningResourceType":"Practice problem","suggestedAnswer":[],"text":"In the given figure, ∠ P Q R = 100 ° , where P , Q and R are points on a circle with centre O . Find ∠ O P R .
\n\n
\n"},"name":"Quiz on Circles","typicalAgeRange":"10-17","url":"https://www.embibe.com/questions/In-the-given-figure%2C-%E2%88%A0PQR%3D100%C2%B0%2C-where-P%2C%C2%A0Q-and-R-are-points-on-a-circle-with-centre-O.-Find-%E2%88%A0OPR.%0A%0A/EM5780111"}EXERCISE Author: Jharkhand Board
Jharkhand Board Mathematics Solutions for EXERCISE Simple step-by-step solutions to EXERCISE questions of Circles from MATHEMATICS: Class IX. Also get 3D topic explainers, cheat sheets, and unlimited doubts solving on EMBIBE.
Questions from EXERCISE with Hints & Solutions MEDIUM
9th Jharkhand Board
IMPORTANT
In the given figure, ∠ P Q R = 100 ° , where P , Q and R are points on a circle with centre O . Find ∠ O P R .
EASY
9th Jharkhand Board
IMPORTANT
In the given figure, ∠ A B C = 69 ° , ∠ A C B = 31 ° , find ∠ B D C . {write only numerical value without degrees symbol}
EASY
9th Jharkhand Board
IMPORTANT
In the given figure, A , B , C and D are four points on a circle. A C and B D intersect at a point E such that ∠ B E C = 130 ° and ∠ E C D = 20 ° . If ∠ B A C = k ° , then find the value of k .
MEDIUM
9th Jharkhand Board
IMPORTANT
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
EASY
9th Jharkhand Board
IMPORTANT
Two circles intersect at two points B and C . Through B , two line segments A B D and P B Q are drawn to intersect the circles at A , D and P , Q respectively. Prove that ∠ A C P = ∠ Q C D .
MEDIUM
9th Jharkhand Board
IMPORTANT
If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.
MEDIUM
9th Jharkhand Board
IMPORTANT
A B C and A D C are two right triangles with common hypotenuse A C . Prove that ∠ C A D = ∠ C B D .
MEDIUM
9th Jharkhand Board
IMPORTANT
Prove that a cyclic parallelogram is a rectangle.