Question

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Answer 1

Given: $A$ circle with center $O$

with tangent $XY$ at point of contact $P.$

To Prove: $OP\perp XY$

Proof: Let $Q$ be a point on $XY$ connect $OQ$

Suppose it touches the circle at $R$

Hence,

$OQ>OR$

$OQ>OP$ $(\because OP=OR)$ (radius)

Same will be the case with all other points on the circle

Hence,

We get $OP$ is the smallest line that connects $XY$.