The local council is building a new pool complex. A one hectare ($100\times100$m) block of land has ground suitable for pool-digging, but a local ring road runs through it, leaving only an exact quarter-circle available.
They want a children's wade pool and a deeper adults' pool, both square. They want to utilize all the suitable land they can, so the adults’ pool must reach the suitable land's edge at three of its corners, where the mandatory three lifeguards will be stationed (dots). Half the children's pool's perimeter must touch the edge, where a stand (L-shape) will be built for parents and lifeguards. The pools must meet at their corners so people can go between them easily.
After seeing the architect’s design (below), the grinchy councillors think the children’s pool is too big. "Exactly how many hectares is this pool?" they ask. The architect says their specifications left him no choice.
How many hectares does the children’s pool occupy?
This is a purely geometrical puzzle of my own construction. Do not assume the image is to scale. The setup is flavourtext and non-essential. The puzzle boils down to: Two squares are inscribed in a unit quadrant as shown in this simplified diagram: what is the blue square’s area? While it is simple and straightforward, and has several possible methods of solution (any of which are welcome), and the answer is not unexpected, there is a simple elegant way to solve it, which is (in my opinion) nice enough to qualify it for this site, rather than being just a maths problem.