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

Answer 1

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).