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Class 9

>>Maths

>>Circles

>>Cyclic Quadrilaterals

>>PQR = 100^ ∘ , where P,Q and R are point

Question



∠PQR=100∘, where P,Q and R are points on a circle with centre O. Find ∠OPR

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Solution



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Here, PR is chord

We mark s on major arc of the circle.

∴ PQRS is a cyclic quadrilateral.

So, ∠PQR+∠PSR=180o

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

100+∠PSR=180o

∠PSR=180o−100o

∠PSR=80o

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=2×80o=160o

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=180o (Angle sum property of triangle)

∠OPR+∠OPR+∠POR=180o from (1)

2∠OPR+160=180o

2∠OPR=180o−160

2∠OPR=20

∠OPR=20/2

∴∠OPR=10o.