An equilateral triangle is described on the diagonal of a square. What is the ratio of the area of the triangle to that of the square?

### Concept & Formula Diagonal of a square with side $s$ is $s\sqrt{2}$. Area of an equilateral triangle with side $a$ is $\frac{\sqrt{3}}{4}a^2$. ### Step-by-Step Solution 1. Let the side of the square be $s$. Area of the square $= s^2$. 2. The diagonal of this square is $s\sqrt{2}$. 3. This diagonal is the side of the equilateral triangle. So, triangle side $a = s\sqrt{2}$. 4. Area of the equilateral triangle $= \frac{\sqrt{3}}{4}(s\sqrt{2})^2 = \frac{\sqrt{3}}{4}(2s^2) = \frac{\sqrt{3}}{2}s^2$. 5. The ratio of the area of the triangle to that of the square is $\frac{\frac{\sqrt{3}}{2}s^2}{s^2} = \frac{\sqrt{3}}{2}$. 6. This can be written as $\sqrt{3} : 2$. ### Exam Strategy & Shortcut Assume the side of the square is 1. Its area is 1. Its diagonal is $\sqrt{2}$. The area of the equilateral triangle on the diagonal is $\frac{\sqrt{3}}{4}(\sqrt{2})^2 = \frac{\sqrt{3}}{2}$. The ratio is simply $\frac{\sqrt{3}}{2} : 1 = \sqrt{3} : 2$. ### Common Pitfall Squaring $s\sqrt{2}$ incorrectly to get just $2s$ instead of $2s^2$, leading to wrong area values. ### Final Answer Therefore, the correct answer is **$\sqrt{3} : 2$**.