Two tangents are drawn from an external point P to the circle with centre O which meet the circle at Q and R respectively. Which of the following is true?

$∠ P Q O = ∠ P O Q$

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$∠ Q P O = ∠ R O P$

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$∠ Q P O = 2 ∠ P O Q$

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$∠ Q P O = ∠ O P R$

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Solution

The correct option is D
$∠ Q P O = ∠ O P R$

$Consider △ P Q O and △ P R O$

$O P = O P (Common side)$

$O Q = O R (Radius of the circle)$

$∠ P Q O = ∠ P R O (Tangents are perpendicular to the radius at the point of contact)$

$So, △ P Q O ≅ △ P R O (By RHS congruency criterion)$

$∴ ∠ Q P O = ∠ R P O (Corresponding sides of congruent triangles)$

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