If PQ and PR are tangents to the circle with centre O and radius 4 ...

Question

If PQ and PR are tangents to the circle with centre O and radius 4 cm such that $\angle QPR = 90^\circ$, then the length OP is

Accepted Answer

Step 1: $OQ\perp PQ$ and $OR\perp PR$ (radius $\perp$ tangent), so $\angle OQP=\angle ORP=90^\circ$.

Step 2: In quadrilateral OQPR, angle sum $=360^\circ$:

$$\angle QOR=360^\circ-90^\circ-90^\circ-90^\circ=90^\circ$$

Step 3: Also $OQ=OR=4$ cm, so OQPR is a square of side 4 cm.

Step 4: OP is its diagonal:

$$OP=\sqrt{4^2+4^2}=\sqrt{32}=4\sqrt2\text{ cm}$$

Answer: (B) $4\sqrt{2}$ cm