In the diagram below, the two circles have the same centre. Point $A$ is on the inner circle and point $B$ is on the outer circle. Line segment $A B$ has length 5 and is tangent to the inner circle at $A$. What is the area of the shaded region?

![](https://cdn.mathpix.com/cropped/2024_04_17_2dbee4c747fee2855e37g-2.jpg?height=364&width=387&top_left_y=1260&top_left_x=934)
Let $C$ be the common centre of the two circles, $R$ be the radius of the larger circle, and $r$ be the radius of the smaller circle.

Since $A B$ is tangent to the smaller circle, we have that $A C$ is perpendicular to $A B$.

This means $\triangle A B C$ is right-angled at $A$, so by the Pythagorean theorem and since $A B=5$, we have $A C^{2}+A B^{2}=B C^{2}$ or $r^{2}+5^{2}=R^{2}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_a164435ebfb99ea5f1c4g-12.jpg?height=447&width=464&top_left_y=1831&top_left_x=1470)

The area of the shaded region is equal to the area of the larger circle minus the area of the smaller circle, or $\pi R^{2}-\pi r^{2}=\pi\left(R^{2}-r^{2}\right)$.

Rearranging $r^{2}+25=R^{2}$, we have $R^{2}-r^{2}=25$, so the area of the shaded region is $\pi\left(R^{2}-r^{2}\right)=25 \pi$.