Intersecting Squares

Weekly Problem 32 - 2014
Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?
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Problem

Image
Intersecting Squares
  

Three congruent squares overlap as shown.

The areas of the three overlapping sections are $2\;\mathrm{cm}^2$, $5\;\mathrm{cm}^2$ and $8\;\mathrm{cm}^2$ respectively.

The total area of the non-overlapping parts of the squares is $117\;\mathrm{cm}^2$.

What is the side length of each square?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.