Two congruent square sheets each have a diagonal of 20 cm. They overlap in a square region. If the total area covered is 384 square cm, find the side of the overlap.

Step-by-step solution

1. 1

   Convert the diagonal to an area. Using $A=\tfrac{d^2}{2}$ for a square of diagonal $d$:

   $A=\frac{20^2}{2}=\frac{400}{2}=200\ \text{cm}^2$

   (The formula follows because the diagonal splits the square into two right isosceles triangles with legs equal to the side.)

2. 2

   Total the two sheets.

   $200+200=400\ \text{cm}^2$

3. 3

   Subtract the covered area to find the double-counted part.

   $|A\cap B|=|A|+|B|-|A\cup B|=400-384=16\ \text{cm}^2$

4. 4

   Take the square root, since the overlap is stated to be a square.

   $s=\sqrt{16}=4\ \text{cm}$

5. 5

   Sanity-check the proportions. Each sheet has side $\tfrac{20}{\sqrt2}\approx 14.1$ cm, so a $4$ cm overlap is a small corner-to-corner intersection — consistent with the covered area $384$ being close to the maximum $400$ $\checkmark$.