Question

From an external point Q, the length of tangent to a circle is 12 cm and the distance of Q from the centre of circle is 13 cm. The radius of circle (in cm) is ______.

Options

• 10

• 5

• 12

• 7

Answer 1

From an external point Q, the length of tangent to a circle is 12 cm and the distance of Q from the centre of circle is 13 cm. The radius of circle (in cm) is 5.

Explanation:

To find the radius r, use the tangent–radius theorem:

OQ² = OT² + QT²

Where

OQ = 13 cm  

QT = 12 cm 

OT = r

So,

13² = r² + 12²

169 = r² + 144

r² = 169 − 144

r² = 25

r = 5

Answer 2

From an external point Q, the length of tangent to a circle is 12 cm and the distance of Q from the centre of circle is 13 cm. The radius of circle (in cm) is 5.

Explanation:

Let O be the centre of the circle.

Given that, OQ = 13 cm and PQ = 12 cm

We know that, the radius is perpendicular to the tangent at the point of contact.

∴ OP ⊥ PQ

In ΔOPQ, using pythagoras theorem,

OP² + PQ² = OQ²

⇒ OP² + 12² = 13²

⇒ OP² = 13² – 12²

⇒ OP² = 169 – 144

⇒ OP² = 25

⇒ OP = 5 cm