Question

In the given figure, AB is a diameter of the circle. The length of AB = 5 cm. If O is the centre of the circle and the length of tangent segment AT = 12 cm, determine CT.

Answer 1

Given:

AB is a diameter

AB = 5 cm

Radius = 5/2 = 2.5 cm

AT is a tangent

AT = 12 cm

C lies on the secant BT

Step 1: Use the tangent–secant theorem

From point T outside the circle:

AT² = CT × (CT + CA)

AT = 12, CA = radius = 2.5 cm

144 = CT (CT + 2.5)

Step 2: Form the quadratic

CT² + 2.5 CT − 144 = 0

Multiply by 2 to clear the decimal:

2CT² + 5CT − 288 = 0

Step 3: Factorise

2CT² + 5CT − 288 = 0

(2CT − 21)(CT + 16) = 0

CT = `21/2 = 21/2 = 10.5`

CT = `− 5 + sqrt(25 + 2304)/4`

= `(− 5 + sqrt2329)/4`

But the simplest exact form comes by keeping decimals:

CT = `144/13` cm