89
PX and PY are two tangents drawn from an external point P to a circle with centre O. If $\angle XPY = 120^{\circ}$, then prove that PX + PY = PO.
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Join OX and OY.
$\angle XPY = 120^{\circ} \Rightarrow \angle XPO = 60^{\circ}$
Now, $\cos 60^{\circ} = \frac{PX}{OP} = \frac{1}{2}$
$\Rightarrow 2 PX = OP$
As PX = PY
$\therefore PX + PY = OP$
$\angle XPY = 120^{\circ} \Rightarrow \angle XPO = 60^{\circ}$
Now, $\cos 60^{\circ} = \frac{PX}{OP} = \frac{1}{2}$
$\Rightarrow 2 PX = OP$
As PX = PY
$\therefore PX + PY = OP$