Given:
- $ \displaystyle PA $ is a tangent to the circle at point $ \displaystyle A $.
- $ \displaystyle O $ is the center of the circle.
- $ \displaystyle \angle APO = 30^\circ $.
- $ \displaystyle OA = 2.5 $ cm.
Since the radius is perpendicular to the tangent at the point of contact, we have:
$ \displaystyle \angle OAP = 90^\circ $.
In right-angled triangle $ \displaystyle OAP $, using the cosine function:
$ \displaystyle \cos \angle APO = \frac{OA}{OP} $
Substituting the values:
$ \displaystyle \cos 30^\circ = \frac{2.5}{OP} $
Since $ \displaystyle \cos 30^\circ = \frac{\sqrt{3}}{2} $, we get:
$ \displaystyle \frac{\sqrt{3}}{2} = \frac{2.5}{OP} $
Cross multiply:
$ \displaystyle OP = \frac{2.5 \times 2}{\sqrt{3}} $
$ \displaystyle OP = \frac{5}{\sqrt{3}} $
Rationalizing the denominator:
$ \displaystyle OP = \frac{5\sqrt{3}}{3} $ cm.
Thus, the length of $ \displaystyle OP $ is **$ \displaystyle \frac{5}{\sqrt{3}} $ cm**.