Question

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

Explanation:

Step1: Let the side length of the square be \(s\)

The diagonal length \(d\) of a square is given by \(d = s\sqrt{2}\) (using the Pythagorean theorem \(d^{2}=s^{2}+s^{2}\)).

Step2: Analyze each option

• Option A: Area of a square

The area \(A=s^{2}\). Since \(s = \frac{d}{\sqrt{2}}\), then \(A=\frac{d^{2}}{2}\). This is a quadratic relationship (\(A\) is proportional to \(d^{2}\)), not a direct proportionality (\(y = kx\) form).

• Option B: Perimeter of a square

The perimeter \(P = 4s\). Substituting \(s=\frac{d}{\sqrt{2}}\), we get \(P=\frac{4}{\sqrt{2}}d=2\sqrt{2}d\). This is in the form \(y = kx\) (\(k = 2\sqrt{2}\)), so perimeter is proportional to diagonal.

• Option C: Side length of a square

Since \(s=\frac{d}{\sqrt{2}}\), this is in the form \(y = kx\) (\(k=\frac{1}{\sqrt{2}}\)), so side - length is proportional to diagonal.

Answer:

B. perimeter of a square, C. side length of a square