Question

In the adjoining figure, O is the centre of the circle. AB and CD are two chords of the circle and OQ ⊥ CD, OP ⊥ AB. If AB = 24 cm, OQ = 12 cm, CD = 10 cm.

Find

1. radius of the circle 
2. length of OP.

Answer 1

Given: O is the centre of the circle. AB and CD are chords with OP ⟂ AB and OQ ⟂ CD.   ...(Perpendicular from centre to a chord bisects the chord)

AB = 24 cm

OQ = 12 cm

CD = 10 cm

Step-wise calculation:

1. CD: Since OQ ⟂ CD

`CQ = 1/2 xx CD`

CQ = 5 cm 

By Pythagoras in △OCQ:

r² = OQ² + CQ² 

= 12² + 5²

= 144 + 25

= 169

⇒ `r = sqrt(169)`

⇒ r = 13 cm

2. AB: Since OP ⟂ AB

`AP = 1/2 xx AB`

AP = 12 cm 

Using r = 13 cm in △OAP:

OP² = r² – AP²

= 169 – 144

= 25

⇒ `OP = sqrt(25)`

⇒ OP = 5 cm

Radius of the circle = 13 cm.

Length of OP = 5 cm.