The length of a tangent from a point at distance from the centre of the circle is . Find the radius of the circle.

Question

Homi · Accepted Answer

Understand the Question

The tangent to a circle is perpendicular to the radius through the point of contact, forming a right-angled triangle with the right angle at the point of contact.
The line segment from the centre to the external point is the hypotenuse of this right triangle.
We use the Pythagoras theorem ( $\text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2$ ) to calculate the unknown radius.

Step 1 · Apply Pythagoras Theorem to Find the Radius

Let $O$ be the centre of the circle and $AB$ be the tangent touching the circle at $B$ . Diagram 1

Since the radius is perpendicular to the tangent at the point of contact: $\text{OB} \perp \text{AB} \implies \angle \text{OBA} = 90^\circ$

Given:

By Pythagoras theorem in right-angled $\Delta \text{OBA}$ : $\text{OA}^2 = \text{OB}^2 + \text{AB}^2$

\begin{aligned} \text{OB}^2 + 4^2 &= 5^2 \\ \text{OB}^2 + 16 &= 25 \\ \text{OB}^2 &= 25 - 16 \\ \text{OB}^2 &= 9 \\ \text{OB} &= \sqrt{9} \\ \text{OB} &= 3\text{ cm} \end{aligned}

Answer

$3\text{ cm}$

Common Mistakes

Hypotenuse Identification: Confusing the tangent length with the hypotenuse. The distance from the centre to the external point ( $\text{OA} = 5\text{ cm}$ ) is always opposite the $90^\circ$ angle and hence is the hypotenuse.
Point of Contact Angle: Assuming the right angle is at the centre $O$ or external point $A$ , whereas $\angle \text{OBA} = 90^\circ$ is always at the point of contact $B$ where the radius meets the tangent.

Question 6