In the given figure, $PA$ is a tangent to a circle with centre $O$....

Question

In the given figure, $PA$ is a tangent to a circle with centre $O$. If $OP=10$ cm and $\angle OPA=30^{\circ}$, then the length of $AP$ is : (Here $A$ is the point of contact and $P$ is an external point.)

Accepted Answer

Step 1: The radius is perpendicular to the tangent at the point of contact, so $\angle OAP=90^{\circ}$.
Step 2: In right $	riangle OAP$, $OP=10$ cm is the hypotenuse and $AP$ is the side adjacent to $\angle OPA=30^{\circ}$.
Step 3: Use the cosine ratio.
$$\cos 30^{\circ}=\frac{AP}{OP}$$
Step 4: Substitute $\cos 30^{\circ}=\dfrac{\sqrt{3}}{2}$ and $OP=10$ cm.
$$\frac{\sqrt{3}}{2}=\frac{AP}{10}$$
Step 5: Therefore $AP=5\sqrt{3}$ cm.
Correct option is (D) $5\sqrt{3}$ cm.

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