Above are two identical isosceles right triangles containing two inscribed squares.
In one, a perfect square has been inscribed such that two sides line up with the two legs of the right triangle. In the other, a perfect square has been inscribed such that one side lines up with the hypotenuse of the right triangle.
Is the inscribed square on the left larger or the one on the right?
Solution
The one on the left. You can calculate it with normal area formulas, or you can use this clever shortcut:
Draw auxiliary lines (shown in blue above) such that you divide the squares and remaining triangles into identically-sized right triangles. Now you can simply count how many triangles each square takes up compared to how many triangles the outer right triangle takes up. The left square is 2/4 and the right is 4/9, so the left square must be larger.