Two congruent square sheets each have a diagonal of 14 cm. They overlap so that the common region is a square. If the total area covered is 160 square cm, find the side of the overlap.

Step-by-step solution

1. 1

   Get the area of one sheet from its diagonal. For a square of diagonal $d$, the side is $\tfrac{d}{\sqrt2}$, so the area is

   $A=\left(\frac{d}{\sqrt2}\right)^2=\frac{d^2}{2}$

   With $d=14$:

   $A=\frac{196}{2}=98\ \text{cm}^2$

2. 2

   Add the two sheets, counting the overlap twice. Laying the sheets on top of one another, the sum of their areas is

   $98+98=196\ \text{cm}^2$

   but the region where they overlap has been counted in both sheets.

3. 3

   Use inclusion-exclusion to isolate the overlap. The area actually covered is the union:

   $|A\cup B|=|A|+|B|-|A\cap B|$

   $160=196-|A\cap B|\quad\Longrightarrow\quad|A\cap B|=36\ \text{cm}^2$

4. 4

   Convert the overlap area to a side length. The problem states the overlap is a square, so its side is the square root of its area:

   $s=\sqrt{36}=6\ \text{cm}$

5. 5

   Check the geometry is consistent. Each sheet has side $\tfrac{14}{\sqrt2}\approx 9.9$ cm, and the overlap side $6$ cm is smaller than that, as it must be $\checkmark$. Working backwards: $196-36=160$ $\checkmark$.