Longest Line Across Concentric Circles

Let the radius of the outer ring be $O \text{ cm}$ , the radius of the inner ring be $I \text{ cm}$ , and the required distance be $x \text{ cm}$ .

The formula for the area of a circle $A = \pi r^2$ is applied to give $100\pi = \pi O^2 - \pi I^2$ and so $100 = O^2 - I^2$ .

From the right-angled triangle formed within the diagram, apply Pythagoras' theorem results in $I^2 + (\frac{x}{2})^2 = O^2$ which rearranges to $(\frac{x}{2})^2 = O^2 - I^2$ .

Equating these two equations gives $(\frac{x}{2})^2 = 100$ , thus $x = 20$ . The correct answer is $20 \text{ cm}$ .

Note that at no point did we discover the values of $O$ or $I$ , but it was not necessary to do so.