Question

Two tangent segments PA and PB are drawn to a circle with centre O such that $\angle APB={120}^{\circ }$, Prove that OP = 2 AP

Answer 1

To prove: OP = 2AP
In $\Delta OBP,sin{60}^{\circ }=\frac{OB}{OP}\Rightarrow \frac{\sqrt{3}}{2}=\frac{r}{OP}\Rightarrow OP=\frac{2r}{\sqrt{3}}\dots \dots \left(1\right)$
Also, AP = BP
$\Rightarrow tan{60}^{\circ }=\frac{OB}{BP}\Rightarrow \sqrt{3}=\frac{r}{BP}\Rightarrow BP=\frac{r}{\sqrt{3}}\dots \dots \left(2\right)$
From (1) and (2)
OP = 2BP = 2AP