The isosceles formula

Everything so far was about particular numbers. Now make it general.

Take an isosceles right triangle with equal sides of length a and hypotenuse c. Two such triangles make a square of side a, whose area is a².

The square on the hypotenuse has double that area: the same fact as before. So c² = 2a².

That one relation works both ways: given the equal sides, find the hypotenuse; given the hypotenuse, find the equal sides.

Forwards. Suppose the equal sides are 12. Then c² = 2 × 12² = 2 × 144 = 288, so c = √288. That isn't a whole number, so bound it: 16² = 256 and 17² = 289, and 288 sits just under 289. So the hypotenuse is between 16 and 17, very close to 17.

Backwards. Suppose the hypotenuse is √72. Then c² = (√72)² = 72, and c² = 2a² gives 72 = 2a². Divide by 2: a² = 36, so a = √36 = 6. Each of the other two sides is exactly 6. A messy-looking hypotenuse can hide perfectly tidy sides.