Correct Answer: (B) $10.25 \, \text{cm}$ **Step-by-step Solution:** 1. **Understand the Geometry and Identify Key Properties:** Let the circle have center $O$ and radius $R$. Let $AB$ be a chord of length $10 \, \text{cm}$. Let $M$ be the midpoint of the chord $AB$. The distance from the center $O$ to the chord $AB$ is $OM = 4 \, \text{cm}$. Since $M$ is the midpoint of $AB$, we have: $$ AM = MB = \frac{1}{2} \times AB = \frac{1}{2} \times 10 \, \text{cm} = 5 \, \text{cm} $$ The radius $OA$ is perpendicular to the tangent $PA$ at point $A$. Thus, $\angle OAP = 90^\circ$. Similarly, $OM$ is perpendicular to the chord $AB$. Thus, $\angle OMA = 90^\circ$. 2. **Calculate the Radius of the Circle ($R$):** Consider the right-angled triangle $\triangle OMA$. Using the Pythagorean theorem: $$ OA^2 = OM^2 + AM^2 $$ Substitute the known values: $$ R^2 = 4^2 + 5^2 $$ $$ R^2 = 16 + 25 $$ $$ R^2 = 41 $$ So, the radius of the circle is $R = \sqrt{41} \, \text{cm}$. 3. **Establish Collinearity of $O$, $M$, and $P$:** Tangents from an external point $P$ to a circle are equal in length, so $PA = PB$. The line segment $OP$ bisects the angle $\angle APB$ and also bisects the angle $\angle AOB$. The line segment $OM$ is the perpendicular bisector of the chord $AB$. Since $P$ is equidistant from $A$ and $B$ (as $PA=PB$), and $O$ is equidistant from $A$ and $B$ (as $OA=OB=R$), the line $OP$ must be the perpendicular bisector of $AB$. Therefore, the points $O$, $M$, and $P$ are collinear. 4. **Utilize Similar Triangles:** Now, consider the right-angled triangle $\triangle OMA$ (right-angled at $M$) and $\triangle OAP$ (right-angled at $A$). Both triangles share the angle $\angle AOM$ (which is the same as $\angle AOP$ because $O, M, P$ are collinear). By the Angle-Angle (AA) similarity criterion, $\triangle OMA \sim \triangle OAP$. 5. **Find the Length of $OP$ using Similarity Ratios:** From the similarity $\triangle OMA \sim \triangle OAP$, the ratio of corresponding sides is equal: $$ \frac{OM}{OA} = \frac{OA}{OP} = \frac{AM}{AP} $$ To find $OP$, we use the first two parts of the ratio: $$ \frac{OM}{OA} = \frac{OA}{OP} $$ Substitute the known values: $OM = 4 \, \text{cm}$ $OA = R = \sqrt{41} \, \text{cm}$ $$ \frac{4}{\sqrt{41}} = \frac{\sqrt{41}}{OP} $$ Now, solve for $OP$: $$ OP = \frac{\sqrt{41} \times \sqrt{41}}{4} $$ $$ OP = \frac{41}{4} $$ $$ OP = 10.25 \, \text{cm} $$ Thus, the length of $OP$ is $10.25 \, \text{cm}$. The final answer is $\boxed{\text{10.25 cm}}$.