A rectangle and a quarter circle. What is the area of the rectangle?

Solution

The rectangle area is 168.

First notice the 3-4-5 triangle AFC. It gives us the rectangle diagonal: x²+y²=25².

Next we discern the cyclic quadrilateral AFDC. It has known diagonals and two known sides. The unknown sides are exactly the rectangle sides we need. Now we can use Ptolemy’s Theorem. It gives 15x+20y=20*25.

Solving these two equations gives x=24 and y=7, leading to an area of 168.