Question

If PQ and PR are tangents to the circle with centre O and radius 4 cm such that ∠QPR = 90°, then the length OP is

Options

• 4 cm

• `4sqrt(2)` cm

• 8 cm

• `2sqrt(2)` cm

Answer 1

`bb(4sqrt(2)  cm)`

Explanation:

Given:

Radius of circle (OQ and OR) = 4 cm

∠QPR = 90°

PQ and PR are tangents to the circle.

To find: Length of OP

In quadrilateral OQPR:

∠OQP = 90°   ...(Radius is perpendicular to tangent at point of contact)

∠ORP = 90°   ...(Radius is perpendicular to tangent at point of contact)

∠QPR = 90°   ...(Given)

Sum of angles in a quadrilateral is 360°:

∠QOR = 360° – (90° + 90° + 90°)

∠QOR = 90°

Since all angles are 90° and adjacent sides OQ = OR = 4 cm, OQPR is a square.

In right-angled triangle OQP:

Using Pythagoras theorem:

OP² = OQ² + QP²

Since OQPR is a square, OQ = QP = 4 cm

OP² = 4² + 4²

OP² = 16 + 16

OP² = 32

OP = `sqrt(32)`

OP = `4sqrt(2)` cm