Question

Prove that, tangents drawn to the end points of diameter of the circle are parallel to each other.

Answer 1

Given: $AB$ is the diameter of the circle with center $O.$

Two tangents $PQ$ and $RS$ are drawn at points $B$ and $A$ respectively.

Proof:

The radius is perpendicular to these tangents.

That is, $OA$ $\perp$ $RS$ and $OB$ $\perp$ $PQ$

$\Rightarrow \angle OAR=\angle OAS=\angle OBP=\angle OBQ={90}^o$

$\therefore \angle OAR=\angle OBQ$ (Alternate interior angles)

$\therefore \angle OAS=\angle OBP$ (Alternate interior angles)

Since alternate interior angles are equal, lines $PQ$ and $RS$ are parallel.