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Question

# The pair of tangents AP and AQ drawn from an external point to a circle with centre O are perpendicular to each other and length of each tangent is 5 cm. The radius of the circle is (a) 10 cm (b) 7.5 cm (c) 5cm (d) 2.5 cm

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Solution

## Given: AP and AQ are tangents to the circle with centre O, AP ⊥ AQ and AP = AQ = 5 cm we know that radius of a circle is perpendicular to the tangent at the point of contact ⇒ OP ⊥ AP and OQ ⊥ AQ Also sum of all angles of a quadrilateral is 360° ⇒∠O + ∠P + ∠A + ∠Q = 360° ⇒∠O + 90° + 90° + 90° = 360° ⇒∠O = 360° – 270° = 90° Thus ∠O = ∠P = ∠A = ∠Q = 90° ⇒ OPAQ is a rectangle Since adjacent sides of OPAQ i.e. AP and AQ are equal. Thus OPAQ is a square radius = OP = OQ = AP = AQ = 5 cm Hence, the correct answer is option (c).

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