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Question

In the given figure, PQR=100, where P, Q and R are the points on a circle with centre O. What will be the value of OPR?


A
30
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B
10
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C
20
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D
40
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Solution

The correct option is B 10
Take any point A on the circumcircle of the circle outside of the arc PQR. Join AP and AR.



Consider the cyclic quadrilateral APQR so formed.
Since opposite angles are supplementary in a cyclic quadrilateral,
PAR+PQR=180.

i.e., PAR+100=180
PAR=80

Since POR and PAR are the angles subtended by the arc PQR at the centre of the circle and the remaining part of the circle, we have

POR=2×PAR
=2×80=160.

In POR, we have
OP=OR (radius of the circle).
Since angles opposite to equal sides are equal,
OPR=ORP.
Using angle sum property,
POR+OPR+ORP=180.
160+OPR+OPR=180
2OPR=20
OPR=10

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