Question

In the figure , two tangents PQ and PR are drawn with centre O from an external point P . Prove that ∠QPR=2∠OQR

Answer 1

Step-by-step explanation:

Given that PQ and QR are two tangents drawn to a circle with centre O from an external point P.

To prove: ∠QPR = 2∠OQR

Construction: Join QR, OQ and OR.

Lengths of a tangent drawn from an external point to a circle are equal.  

PQ = QR  ,

Therefore, ΔPQR is an isosceles triangle  .

∠PQR = ∠PRQ

In ΔPQR,

∠PQR + ∠PRQ + ∠QPR = 180°

∠PQR + ∠PQR + ∠QPR = 180°

2∠PQR=180° −∠QPR

$\angle P Q R=\frac{1}{2}\left(180^{\circ}-\angle Q P R\right)$

$\angle P Q R=90^{\circ}-\frac{1}{2} \angle Q P R$

$\frac{1}{2} \angle Q P R=90^{\circ}-\angle P Q R$ …………(1)

Since PQ is perpendicular to PQ.

∠OQP=90°

$\angle O Q R+\angle P Q R=90^{\circ}$

$\angle O Q R=90-\angle P Q R$ …………(2)

$\Rightarrow \angle O Q R=\frac{1}{2} \cdot \angle O Q R$

$\Rightarrow 2 \cdot \angle O Q R=\angle Q P R$

$\therefore \angle Q P R=2 \angle O Q R$

Hence it is proved.

To learn more...

1. In the adjoining figure,two tangents PQ and PR are drawn to circle

with Centre O from an external point P.

Prove That ∠QPR = 2∠OQR

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2. In the given figure two tangents PQ and PR are drawn to a circle with center O from an external point P prove that angle QPR=2angle 0QR.

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