Question

In the adjoining figure, O is the centre ofa circle with a diameter AB. If ∠OMC = 90°, CD = 6 cm, BM = 1 cm, find the radius of the circle.

Answer 1

Given:

O is centre, AB is diameter, CD is a horizontal chord with midpoint M on the diameter.

CD = 6 cm, so CM = 3 cm.

BM = 1 cm.

Step-wise calculation:

1. Let the radius be r.

The distance OM centre to midpoint M of chord equals OB – BM = r – 1.

2. In right triangle OMC

OC is radius r, CM = 3, OM = r – 1:

r² = 3² + (r – 1)²

3. Expand and simplify:

r² = 9 + (r² – 2r + 1)

r² cancels on both sides

⇒ 0 = 10 – 2r

2r = 10

⇒ r = 5

The radius of the circle is 5 cm.