Question

In the given figure, O is the centre of the circle and B is a point of contact. seg OE ⊥ seg AD, AB = 12, AC = 8, find (1) AD (2) DC (3) DE.

Answer 1

(1) Line AB is the tangent at point B and seg AD is the secant.    ...(Given)

∴ By the tangent secant segments theorem,

∴ AC × AD = AB²

∴ 8 × AD = 12²

∴ 8 × AD = 144

∴ AD = `144/8`

∴ AD = 18 units

(2) AD = AC + DC     ... [A – C – D]

∴ 18 = 8 + DC

∴ DC = 18 – 8

∴ DC = 10 units

(3) seg OE ⊥ seg AD     ...[Given]

i.e. seg OE ⊥ seg CD     ...[A – C – D]

∴ DE = `1/2 × "DC"`    ...(Perpendicular drawn from the centre of the circle to the chord bisects the chord)

∴ DE = `1/2 × 10`

∴ DE = 5 units