Question

Theorem: Tangent segments drawn from an external point to a circle are congruent

Answer 1

Given: $A$ is the centre of the circle.
Tangents through external point $D$ touch the circle at the points $P$ and $Q$.
To prove: seg DP $\cong$ seg $DQ$
Construction: Draw seg AP and seg AQ.

Proof:
In $△PAD$ and $△QAD$,
seg $PA\cong [$ segQA] [Radii of the same circle]
seg $AD\cong \mathrm{seg}AD$ [Common side]
$\angle APD=\angle AQD={90}^{\circ }$ [Tangent theorem]
$\therefore △PAD=△QAD[$ By Hypotenuse side test]
$\therefore$ seg $DP=\mathrm{seg}DQ$ [c.s.c.t]