Question

In the following figure, PB is a tangent to the circle with centre O at B. AB is a chord of the circle of length 24 cm and at a distance of 5 cm from the centre of the circle. If the length PB of the tangent is 20 cm, find the length of OP.

Answer 1

Given:

AB = 24 cm (chord), distance OM (from centre O to chord AB) = 5 cm.

PB (tangent from P to circle at B) = 20 cm.

Let O be centre, M the midpoint of AB, and r = radius = OB.

Step-wise calculation:

1. Since OM is perpendicular to chord AB, M is the midpoint of AB.

So AM = MB = `24/2` = 12 cm.

In right triangle OMB:

r² = OM² + MB² 

= 5² + 12²

= 25 + 144

= 169

⇒ r = 13 cm

2. For an external point P, the power-of-a-point (tangent) relation gives:

PB² = PO² – r² 

Substitute PB = 20 and r = 13:

20² = PO² – 13²

400 = PO² – 169

PO² = 400 + 169 = 569

PO = `sqrt(569)` ≈ 23.85 cm