Two identical square sheets of paper each have a diagonal of 24 cm. They are laid on top of each other so that the overlapping region is itself a square with side AB. If the total area covered by the combined figure is 384 square cm, find the length of AB.

Step-by-step solution

1. 1

   Use the diagonal formula for a square's area. With $d=s\sqrt2$ we get $s^2=\tfrac{d^2}{2}$, so

   $\text{Area}=\frac{1}{2}(24)^2=\frac{576}{2}=288\ \text{cm}^2$

   per sheet. Note the area scales with $d^2$: doubling the diagonal from $12$ to $24$ would quadruple the area.

2. 2

   Apply inclusion-exclusion. The covered area counts the shared region once, not twice:

   $\text{Covered}=\text{Area}_1+\text{Area}_2-\text{Overlap}$

3. 3

   Solve for the overlap. Total paper is $288+288=576\ \text{cm}^2$, but only $384\ \text{cm}^2$ is covered:

   $\text{Overlap}=576-384=192\ \text{cm}^2$

4. 4

   Take the square root, since the overlap is a square.

   $AB=\sqrt{192}\ \text{cm}$

5. 5

   Simplify the radical. Pull out the largest perfect-square factor: $192=64\times3$, and $\sqrt{64}=8$, so

   $AB=8\sqrt3\approx13.86\ \text{cm}$

   Unlike the $\sqrt{78}$ case of the $18$ cm version, this radical simplifies cleanly.

6. 6

   Check plausibility. Each sheet has side $\sqrt{288}\approx16.97$ cm, and $13.86<16.97$ ✓, so the overlap fits inside a sheet. The overlap is $\tfrac{192}{288}=\tfrac{2}{3}$ of each sheet — a large but perfectly possible overlap ✓.