Question

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Answer 1

STEP 1 : Draw the diagram

Let us draw a circle with centre $O$ and diameter as $AB$.

Let $PQ$ be the tangent at point $A$ and $RS$ be the tangent at point $B$

STEP 2 : Proving that the tangents drawn from point $A$ and $B$ are parallel

We know that $PQ$ is a tangent at point $A$. Therefore,

$OA\perp PQ$ (Tangent at any point on the circle is perpendicular to the radius through point of contact)

$\angle OAP=90°$ $...\left(1\right)$

Similarly, we know that $RS$ is a tangent at point $B$. Thus,

$OB\perp RS$ (Tangent at any point on the circle is perpendicular to the radius through point of contact)

$\angle OBS=90°$ $...\left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$ we get,

$\angle OAP=\angle OBS=90°$

$\Rightarrow \angle BAP=\angle ABS$

Since, $AB$ is a transversal for lines $PQ$ and $RS$ and $\angle BAP=\angle ABS$

i.e. If alternate angles are equal then the lines are parallel

Therefore, $PQ\parallel RS$

Hence, the tangents drawn at the ends of a diameter of a circle are parallel.