Question

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Answer 1

To prove : $\angle 1=\angle 2$.
Let $RQ$ be a chord of the circle.
Tangents are drawn at the point $R$ and $Q.$

Let $P$ be another point on the circle, then.
Join $PQ$ and $PR.$
Since, at point $Q$ , there is a tangent.
$\therefore$ $\angle 2$ = $\angle P$ [Angles in alternate segments are equal]
Since at point R, there is a tangent.
$\therefore$ $\angle 1$ = $\angle P$ [Angles in alternate segments are equal]
$\therefore$ $\angle 1$ = $\angle 2$ = $\angle P$
Hence, tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Alternative Solution:

Extend the two tangents to maat at a point $O$.

Here , $RQ$ is the chord from whose ends the tangents are drawn.

The Tangents $QO$ and $PO$ meet at $O.$ We need to prove
$\angle OQR=\angle ORQ$.

We know, $OQ=OR$
(Tangents drawn from an external point to the circle are equal)

$\therefore$ $\angle OQR=\angle ORQ$ (Angles opposite to opposite sides are equal)
Hence, tangents drawn at the ends of a chord of a circle make equal angles with the chord.