Question

Prove that tangent segments drawn from an external point to a circle are congruent.

Answer 1

Solution:
Given: $O$ is centre of the circle, $D$ is the external point. DP and DQ are the tangent of the circle.
To prove: $DP=DQ$
Construction: Draw AP and AQ.
Proof: In $△PAD$ and $△QAD$,
$AP=AQ\dots$ (Radii of the same circle)
$\angle APD=\angle AQD$... (tangent theorem)
$AD=AD\dots$ (common)
$\therefore △PAD\cong △QAD\dots$ (By SAS test)
$\therefore DP=DQ\dots$ (CPCT)
Hence, proved.