Question

Chords AB and CD of a circle are parallel to each other and lie on the opposite sides of the centre of the circle. If AB = 48 cm, CD = 36 cm and the distance between the chords is 42 cm, find the radius of the circle.

Answer 1

Given: AB = 48 cm, CD = 36 cm, distance between the chords = 42 cm chords parallel and on opposite sides of centre.

Step-wise calculation:

1. Let O be the centre.

Drop perpendiculars OP ⟂ AB and OQ ⟂ CD at their midpoints.

Then AP = 24 cm and CQ = 18 cm.

Let OP = x and OQ = y.

2. Because the chords lie on opposite sides of O.

x + y = 42

3. Radius r satisfies: 

r² = x² + 24²

And r² = y² + 18² 

Equate: x² + 576 = y² + 324

⇒ x² – y² = –252 

⇒ (x – y)(x + y) = –252

4. Substitute x + y = 42: 

(x – y) × 42 = –252 

⇒ x – y = –6

5. Solve x + y = 42 and x – y = –6: 

2x = 36

⇒ x = 18 cm

Then y = 24 cm.

6. Compute r: 

r² = x² + 24²

= 18² + 576

= 324 + 576

= 900

⇒ r = 30 cm

Radius of the circle = 30 cm.