Question

In the adjoining figure, a circle with centre O is shown in which AB is the diameter which bisects the chord CD at point M. If CD = 16 cm, BM = 4 cm, find the diameter of the circle.

Answer 1

Given:

AB is a diameter through centre O and it bisects chord CD at M, so OM is the perpendicular from the centre to chord CD, hence OM ⟂ CD.

CD = 16 cm

So, CM = `1/2` × CD = 8 cm

BM = 4 cm

Step-wise calculation:

1. Let the radius be r.

Place O at the origin on AB.

Then B is at x = +r and M is at x = r – BM = r – 4.

So, OM = r – 4.

2. In right triangle OCM right angle at M. 

Apply Pythagoras:

OC² = OM² + CM²

r² = (r – 4)² + 8²

3. Expand and simplify:

r² = r² – 8r + 16 + 64

0 = –8r + 80 

8r = 80

r = 10 cm

Diameter = 2r = 20 cm.