An equilateral triangle is drawn on one side of a square, and a second equilateral triangle is drawn on a diagonal of the same square.
Find the ratio of the area of the first equilateral triangle to the area of the second.

Let the square have side a; its diagonal is $a\sqrt{2}$. All equilateral triangles are similar to one another, so their areas are in the ratio of the squares of the sides they stand on: $\frac{a^2}{(a\sqrt{2}{)}^2}=\frac{a^2}{2a^2}=\frac{1}{2}$, i.e. 1 : 2. The options 1 : √2 and √2 : 1 stop at the ratio of the two lengths, and 1 : 4 comes from taking the diagonal to be $2a$ instead of $a\sqrt{2}$.