Problem 2 Select all quantities that are proportional to the diagonal length of a square. A. Area of a square B. Perimeter of a square C. Side length of a square

Let the side length of the square be \( s \). 1. The diagonal length \( d \) of the square is given by \[ d = s\sqrt{2}. \] 2. **Area of a square:** The area is \[ A = s^2. \] Express \( A \) in terms of \( d \): \[ s = \frac{d}{\sqrt{2}} \quad \Longrightarrow \quad A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}. \] This shows that the area is proportional to \( d^2 \), not directly proportional to \( d \). 3. **Perimeter of a square:** The perimeter is \[ P = 4s. \] Substitute \( s \) in terms of \( d \): \[ P = 4\left(\frac{d}{\sqrt{2}}\right) = \frac{4d}{\sqrt{2}} = 2\sqrt{2}\,d. \] Thus, the perimeter is directly proportional to \( d \). 4. **Side length of a square:** We already have \[ s = \frac{d}{\sqrt{2}}, \] which shows that the side length is directly proportional to \( d \). Hence, the quantities proportional to the diagonal length of a square are the perimeter and the side length. The correct selections are **B. Perimeter of a square** and **C. Side length of a square**.