Question

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. What is the radius of the circle?

Answer 1

Let the radius of the circle is r cm.

Let the point of contact of tangent and the circle is point P.

Therefore, the length of the tangent to a circle from a point Q is PQ = 24 cm.

And the distance of Q from the centre is OQ = 25 cm.

And the radius of the circle is OP = r cm.

Since, tangent is perpendicular to the line joining its point of contact and the centre of the circle, therefore, QP⏊ OP. Hence, \(\angle\)OPQ is a right angle in triangle ∆OPQ.

Therefore, OQ² = OP² + PQ². (By Pythagoras theorem in triangle ∆OPQ.)

⇒ OP² = OQ² − PQ² = 252 − 242 = 625 − 576 = 49.

⇒ OP = r = \(\sqrt{49}\) = 7cm.

Hence, the radius of the circle is r = 7cm.