Question

In the figure, $O$ is the center of the circle, $AB$ and $AC$ are tangents. Radius of the circle is $r$ and $l\left(AB\right)=r$, Prove that $ABOC$ is a square.

Answer 1

As length of tangents from

a point (A)to circle are equal

$\Rightarrow$ AB=AC=r

and radius$\perp$ tangent$\Rightarrow \angle B=\angle c={90}^{\circ }$

In $△AOB,AB=OB=r\Rightarrow \angle AOB=\angle BAO$

In $△AOB,$ sum of angles $={180}^{\circ }\Rightarrow \angle AOB=\angle BAO={45}^{\circ }$

similarly, $\angle AOC=\angle OAC={45}^{\circ }\Rightarrow \angle BOC={90}^{\circ }$, $\angle BAC={90}^{\circ }\Rightarrow$ All angle are ${90}^{\circ }$ and all sides are equal $⟹$ $ABOC$ is a square