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 ∠QPR and ∠OPQ is ______.

2 : 1

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1 : 1

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4 : 1

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1 : 2

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Solution

The correct option is A 2 : 1

Join 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,
$\frac{∠ Q P R}{∠ O P Q} = \frac{∠ Q P O + ∠ O P R}{∠ O P Q} = \frac{2 \times ∠ O P Q}{∠ O P Q} = \frac{2}{1}$

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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 ∠QPR and ∠OPQ is ______.

The correct option is A 2 : 1 Join OQ and OR. Consider ΔOPQ and ΔOPR OP = OP (common side) OQ = OR (radii) ∠OQP=∠ORP=90∘ (Tangent is perpendicular to the radius) Hence, ΔOQP≅ΔORP (RHS congruency) Hence, ∠OPQ=∠OPR (CPCT) So, ∠QPR∠OPQ=∠QPO+∠OPR∠OPQ=2×∠OPQ∠OPQ=21