Question

Short Answer Type
26. In the following figure, O is the centre of the circle. Prove that:
POR = 2 (PRO + QPR).​

Answer 1

Explanation:

To prove: POR = 2 ( PRO+QPR )

Construction: Join QR, OQ and OR.

Proof: We know that lengths of a tangent drawn from an external point to a circle are equal.

PQ=QR

ΔPQR is an isosceles triangle

∠PQR=∠PRQ

In ΔPQR

∠PQR+∠PRQ+∠QPR=180

o

∠PQR+∠PQR+∠QPR=180

o

2.∠PQR=180

o

−∠QPR

∠PQR=

2

1

(180

o

−∠QPR)

∠PQR=90

o

−

2

1

∠QPR

2

1

∠QPR=90

o

−∠PQR …………(1)

Since PQ is perpendicular to PQ.

∠OQP=90

o

∠OQR+∠PQR=90

o

∠OQR=90−∠PQR ………..(2)

⇒∠OQR=

2

1

⋅∠OQR

⇒2.∠OQR=∠QPR

∴∠QPR=2∠OQR

Hence, it proved.