Question

determine if there is a proportional relationship between the length of the diagonals and the perimeters of the squares.
square (cm) side lengths (cm) perimeter (cm)
a 2.7
b 3.3
c 4
d 4.5

Explanation:

Step1: Recall perimeter formula for square

The perimeter $P$ of a square with side - length $s$ is $P = 4s$.

Step2: Calculate perimeter of square A

For square A with $s = 2.7$ cm, $P_A=4\times2.7 = 10.8$ cm.

Step3: Calculate perimeter of square B

For square B with $s = 3.3$ cm, $P_B=4\times3.3 = 13.2$ cm.

Step4: Calculate perimeter of square C

For square C with $s = 4$ cm, $P_C=4\times4 = 16$ cm.

Step5: Calculate perimeter of square D

For square D with $s = 4.5$ cm, $P_D=4\times4.5 = 18$ cm.

Step6: Recall diagonal formula for square

The diagonal $d$ of a square with side - length $s$ is $d=\sqrt{2}s$. Let's find the ratios of diagonal to perimeter. The ratio of the diagonal $d$ to the perimeter $P$ is $\frac{d}{P}=\frac{\sqrt{2}s}{4s}=\frac{\sqrt{2}}{4}$, which is a constant.

Answer:

Yes, there is a proportional relationship between the length of the diagonals and the perimeters of the squares. The perimeters of squares A, B, C, and D are 10.8 cm, 13.2 cm, 16 cm, and 18 cm respectively.