From a point $P$, two tangents $PA$ and $PB$ are drawn to a circle with centre $O$. If $OP=13\text{ cm}$ and the radius of the circle is $5\text{ cm}$, what is the length of each tangent segment $PA$ (or $PB$)?

Step-by-step solution

1. Join $O$ to $A$; $OA=5\text{ cm}$ (radius) and $OP=13\text{ cm}$ (given).
2. Since a radius is perpendicular to the tangent at the point of contact, $△OAP$ is right‑angled at $A$.
3. Apply Pythagoras: $P{A}^2=O{P}^2-O{A}^2=13^2-5^2=169-25=144$.
4. Take the positive root: $PA=\sqrt{144}=12\text{ cm}$.

Concept tested: Length of tangent from external point

Published 20 August 2026 · Last updated 16 September 2026 · How this solution was prepared and checked