[Math] How to cut a square in half

Question

I have a square that's $10\mathrm{m} \times 10\mathrm{m}$. I want to cut it in half so that I have a square with half the area. But if I cut it from top to bottom or left to right, I don't get a square, I get a rectangle!

I know the area of the small square is supposed to be $50\mathrm{m}^{2}$, so I can use my calculator to find out how long a side should be: it's $7.07106781\mathrm{m}$. But my teacher said I should be able to do this without a calculator. How am I supposed to get that number by hand?

Answer 1

Does this give you any ideas?