Two congruent square sheets each have a diagonal of 28 cm. They overlap in a square region. If the total area covered is 720 square cm, find the side of the overlap.

Step-by-step solution

1. 1

   Find one sheet's area from the diagonal.

   $A=\frac{d^2}{2}=\frac{28^2}{2}=\frac{784}{2}=392\ \text{cm}^2$

2. 2

   Add both sheets. This counts the overlap region twice:

   $392+392=784\ \text{cm}^2$

3. 3

   Apply inclusion-exclusion with the given covered area.

   $|A\cap B|=784-720=64\ \text{cm}^2$

4. 4

   Extract the side of the square overlap.

   $s=\sqrt{64}=8\ \text{cm}$

5. 5

   Check, and note when this type of problem breaks. Each sheet's side is $\tfrac{28}{\sqrt2}\approx 19.8$ cm, comfortably larger than $8$ cm $\checkmark$, and $784-64=720$ $\checkmark$. The method only yields a whole-number side when the difference $\tfrac{d^2}{2}\cdot 2-|A\cup B|$ happens to be a perfect square: for instance $d=24$ with covered area $528$ gives an overlap of $48$, whose square root is irrational, so that combination cannot have an integer answer.