Question

Prove that a line drawn through the end point of a radius and perpendicular to it is a tangent to the circle.

Answer 1

Step1 : Drawing a diagram

Here, $OP$ is the radius and the line $APB$ is perpendicular to $OP$. Let $Q$ be any point on the line $APB$.

Step 2: Proof

Since, $OP\perp AB$, therefore,

$OP<OQ$ (Distance of point from a line is the shortest distance between any point on the line and the given point)

Therefore,

$OP$ is shorter than any other line segment connecting line $APB$ and $O$.

Since $OP$ is the radius of the circle therefore, $Q$ lies outside the circle that means every point on line $APB$ lies outside the circle except $P$.

This implies line $APB$ meets the circle only and only at point $P$.

Therefore, $APB$ is tangent to the given circle.