In the given figure, PA and PB are two tangents drawn to the circle with centre O and radius 5 cm. If $\angle APB = 60^\circ$, then the length of PA is :

Step 1: Identify properties of tangents and angles

When a tangent is drawn to a circle, the radius drawn to the point of tangency is perpendicular to the tangent. Therefore, angles $\angle OAP$ and $\angle OBP$ are both $90^\circ$.

Step 2: Find angle AOP using quadrilateral properties

Consider the quadrilateral OAPB. The sum of angles in a quadrilateral is $360^\circ$. Since $\angle OAP = 90^\circ$, $\angle OBP = 90^\circ$, and $\angle APB = 60^\circ$, we can find $\angle AOB = 360^\circ - (90^\circ + 90^\circ + 60^\circ) = 360^\circ - 240^\circ = 120^\circ$. Alternatively, the angle between the two tangents from an external point and the angle subtended by the chord joining the points of contact at the center are supplementary, so $\angle AOB = 180^\circ - 60^\circ = 120^\circ$.

Step 3: Determine angle APO

The line segment PO bisects $\angle APB$ and $\angle AOB$. Therefore, $\angle APO$ is half of $\angle APB$.

Step 4: Use trigonometry in right-angled triangle OAP

In the right-angled triangle $\triangle OAP$, we know the angle $\angle APO = 30^\circ$ and the side $OA = 5\text{ cm}$ (radius). We can use the tangent function, which is the ratio of the opposite side to the adjacent side.

Step 5: Calculate the length of PA

Substitute the value of $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ into the equation and solve for PA.

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