Question

Prove that, "the lengths of the two tangent segments to a circle drawn from an external point are equal".

Answer 1

Given: A circle with centre $O$, an external point $P$ of the circle. The two tangents through the point $P$ touches the circle at the points $A$ and $B$.

To prove: $PA=PB$

Construction: Draw seg $OA$, seg $OB$ and seg $OP$.

Proof: $\angle PAO=\angle PBO$ ... (Radius is perpendicular to tangent).

In the right angled $\Delta PAO$ and the right angled $\Delta PBO$,

1. $segOA\cong segOB$ ... (Radii of same circle),
   
2. hypotenuse $PO\cong$ hypotenuse $PO$ ... (common side)
   
3. $\Delta PAO\cong \Delta PBO$ ... (Hypotenuse side theorem)
   

$\therefore segPA\cong segPB$

$\therefore PA=PB$