From an external point P, two tangents are drawn that touch the circle at points Q and R. The centre of the circle is O. Points O and P are joined. The ratio of $∠ O P R a n d ∠ O P Q$ is ______.

1 : 1

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3 : 1

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4 : 1

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2 : 1

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Solution

The correct option is A
1 : 1

Join the points OP, OQ and OR.

Consider $Δ O P Q a n d Δ O P R$
OP = OP (common side)
OQ = OR (radii)
$∠ O Q P = ∠ O R P = 90^{\circ}$ (Tangent is perpendicular to the radius)
Hence,
$Δ O Q P ≅ Δ O R P$ (RHS congruency)

Hence,
$∠ O P Q = ∠ O P R (C P C T)$

So,
$\frac{∠ O P R}{∠ O P Q} = \frac{1}{1}$

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