A point P is 13 cm from the center of a circle. The length of the tangent drawn from P to the circle is 12 cm. Find the radius of the circle.

Correct option is A

Given:
Distance of point P from the center of the circle, OP = 13 cm
Length of the tangent from P  to the circle, PT = 12 cm
Solution:
Let O  be the center of the circle, T  be the point of contact of the tangent, and OT = r  (radius of the circle).
Since the tangent is perpendicular to the radius at the point of contact,  $$\triangle OTP$$​  is a right-angled triangle with $$\angle OTP = 90^\circ .$$​​
Apply the Pythagoras theorem:

$$OP^2 = OT^2 + PT^2$$​​
​$$13^2 = r^2 + 12^2$$​
​$$169 = r^2 + 144$$​​
$$r^2 = 169 - 144$$​​
$$r^2 = 25$$​​
$$r = \sqrt{25} = 5 \, \text{cm}$$​​
Alternate Method:
Recognize that the tangent, radius, and line joining the external point to the center form a right-angled triangle.

Use the formula for the length of the tangent from an external point to a circle:

$$\text{Length of tangent} = \sqrt{OP^2 - r^2}$$​​
​$$12 = \sqrt{13^2 - r^2}$$​​
$$12 = \sqrt{169 - r^2}$$​​
​​$$r^2 = 169 - 144 = 25 \\ \ \\ r = \sqrt{25} = 5 \, \text{cm}$$​​