Question

Angle PQR=100°,whare P,Q and R are points on a circle O.Find angle OPR​

Answer 1

Answer:

10 °

Explanation:

Here, PR is chord

We mark s on major arc of the circle.

∴ PQRS is a cyclic quadrilateral.

So, ∠PQR+∠PSR=180°

[Sum of opposite angles of a cyclic quadrilateral is 180°]

100+∠PSR=180°

∠PSR=180° −100

∠PSR=80°

Arc PQR subtends ∠PQR at centre of a circle.

And ∠PSR on point s.

So, ∠POR=2∠PSR

[Angle subtended by arc at the centre is double the angle subtended by it any other point]

∠POR=2×80°=160°

Now,

In ΔOPR,

OP=OR[Radii of same circle are equal]

∴∠OPR=∠ORP [opp. angles to equal sides are equal] ………………..(1)

Also in ΔOPR,

∠OPR+∠ORP+∠POR=180°

(Angle sum property of triangle)

∠OPR+∠OPR+∠POR=180°

from (1)

2∠OPR+160=180°

2∠OPR=180° −160°

2∠OPR=20°

∠OPR=20/2

∴∠OPR=10°

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