Question

If a tangent $PQ$ at a point $P$ of a circle of radius $5 \,\text{cm}$ meets a line through the centre $O$ at a point $Q$ so that $OQ = 12 \,\text{cm}$, then length of $PQ$ will be:
 

• A. $8.5 \,\text{cm}$
• B. $\sqrt{119} \,\text{cm}$
• C. $12 \,\text{cm}$
• D. $13 \,\text{cm}$

Hint:

In circle tangent problems, always look for right-angled triangles involving radius and tangent. Pythagoras theorem is the key.

Answer 1

The Correct Option is B

Step 1: Understanding the geometry
- A tangent at point $P$ of a circle is perpendicular to the radius $OP$.
- Thus, $\triangle OPQ$ is a right-angled triangle with right angle at $P$.
- Here, $OP = 5 \,\text{cm}$ (radius) and $OQ = 12 \,\text{cm}$.

Step 2: Apply Pythagoras theorem
\[ OQ^2 = OP^2 + PQ^2 \] Substitute values: \[ 12^2 = 5^2 + PQ^2 \] \[ 144 = 25 + PQ^2 \] \[ PQ^2 = 144 - 25 = 119 \] \[ PQ = \sqrt{119} \,\text{cm} \]

Step 3: Conclusion
The length of $PQ$ is $\sqrt{119} \,\text{cm}$.
The correct answer is option (B).