Question

determine if there is a proportional relationship between the length of the diagonals and the area of the squares. square (cm) side lengths (cm) area (cm²) a 1.1 b 2.2 c 2.4 d 3.1

Explanation:

Step1: Recall area formula for square

The area formula of a square is $A = s^2$, where $s$ is the side - length of the square.

Step2: Calculate area of square A

For square A with $s = 1.1$ cm, $A_A=1.1^2 = 1.21$ $cm^2$.

Step3: Calculate area of square B

For square B with $s = 2.2$ cm, $A_B=2.2^2 = 4.84$ $cm^2$.

Step4: Calculate area of square C

For square C with $s = 2.4$ cm, $A_C=2.4^2 = 5.76$ $cm^2$.

Step5: Calculate area of square D

For square D with $s = 3.1$ cm, $A_D=3.1^2 = 9.61$ $cm^2$.

Step6: Recall diagonal formula for square

The diagonal of a square $d=\sqrt{2}s$.

Step7: Check proportionality

If two variables $x$ and $y$ are proportional, then $\frac{y}{x}=k$ (constant). Let $x$ be the diagonal length $d = \sqrt{2}s$ and $y$ be the area $A=s^2$. Then $\frac{A}{d}=\frac{s^2}{\sqrt{2}s}=\frac{s}{\sqrt{2}}$, which is not a constant as it depends on $s$. So, there is no proportional relationship between the length of the diagonals and the area of the squares.

Answer:

There is no proportional relationship between the length of the diagonals and the area of the squares. The areas of squares A, B, C, D are 1.21 $cm^2$, 4.84 $cm^2$, 5.76 $cm^2$, 9.61 $cm^2$ respectively.