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 angles OPR and OPQ is:

1:1

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2:1

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3:1

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4:1

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Solution

The correct option is A 1:1

Join the points OP,OQ and OR
Now, in triangles OPQ and OPR,
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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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 angles OPR and OPQ is:

The correct option is A 1:1 Join the points OP,OQ and OR Now, in triangles OPQ and OPR, 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, ∠OPR∠OPQ=11