Question

In the adjoining figure, O is the centre of the circle and B is a point of contact. Seg OE ⊥ seg AD, AB = 12, AC = 8, find 

i. AD 

ii. DC 

iii. DE

Answer 1

i. Line AB is the tangent at point B and seg AD is the secant. [Given] 

∴ AC × AD = AB² [Tangent secant segments theorem] 

∴ 8 × AD = 122 

∴ 8 × AD = 144

∴ AD = 144/8

∴ AD = 18 units 

ii. AD = AC + DC [A – C – D] 

∴ 18 = 8 + DC 

∴ DC = 18 – 8 

∴ DC = 10 units 

iii. 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] 

= (1/2) × 10 

∴ DE = 5 units