Question

In figure, OQ : PQ = 3 : 4 and perimeter of ΔPDQ = 60 cm. Determine PQ, QR and OP.

Answer 1

Given OQ: PQ = 3 : 4

Let OQ = 3x PQ = 4x

OP = y

∠OQP = 90° [since at point of contact, tangent is perpendicular to radius]

In ΔOQP, by Pythagoras theorem

𝑂𝑃² = 𝑂𝑄² + 𝑄𝑃²

⇒ 𝑦² = (3𝑥)² + (4𝑥)²

⇒ 𝑦² = 9𝑥² + 16𝑥² = 25𝑥²

⇒ 𝑦 = `sqrt(25x^2)` = 5𝑥

Perimeter = OQ + PQ + OP = 3x + 4x + 5x = 12x

According to problem perimeter = 60

∴ 12x = 60

x =`60/12`

= 5𝑐𝑚

OQ = 3 × 5 = 15𝑐𝑚

PQ = 4 × 5 = 20 𝑐𝑚

OP = 5 × 5 = 25𝑐𝑚