About circles 2

a) O is the centre of the circle since it is the point of intersection between 2 diameters.

Therefore, OB = OE = 8 cm and hence △OBE is an equiliateral triangle.

Hence ∠BOE = 60

Join OA, then ∠AOE = (1/2) x ∠BOE = 30 since arc AB = arc AE

∠ACE = (1/2) x ∠AOE = 15 (∠ at circum is half of ∠ at centre)

b) Join AE, then ∠CAE = 90 (∠ in semi-circle)

Thus △CAE is a right-angled triangle with:

AC/CE = cos ∠AOE

AC = 16 cos 15 = 15.5 cm