Question

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

Explanation:

Step1: Recall area formula for square

The area of a square with side - length $s$ is $A = s^{2}$.

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 $d$ of a square with side - length $s$ is $d=\sqrt{2}s$. Then $s = \frac{d}{\sqrt{2}}$, and the area $A=\frac{d^{2}}{2}$.

Step7: Check proportionality

If $A=\frac{d^{2}}{2}$, then $\frac{A}{d^{2}}=\frac{1}{2}$. For two squares with diagonals $d_1$ and $d_2$ and areas $A_1$ and $A_2$, $\frac{A_1}{d_1^{2}}=\frac{A_2}{d_2^{2}}=\frac{1}{2}$. So there is a proportional relationship between the length of the diagonals and the areas of the squares.

Answer:

Square A area: $1.21$ $cm^{2}$, Square B area: $4.84$ $cm^{2}$, Square C area: $5.76$ $cm^{2}$, Square D area: $9.61$ $cm^{2}$. There is a proportional relationship between the length of the diagonals and the areas of the squares.