Questions: Determine if there is a constant of proportionality for the lengths of the diagonals and the perimeters of the squares. (1) Does it look like the table is showing a constant of proportionality for the relationship between the diagonals and the perimeters? In other words, are there the same number of centimeters in the perimeter for every 1 centimeter in the diagonal? (1) Square Diagonal Perimeter Perimeter ÷ Diagonal ------------ A 3 8.4 2.8 B 4.5 12.8 2.8 C 5.7 16 2.8 D 7.6 21 More 2.8

Determine if there is a constant of proportionality for the lengths of the diagonals and the perimeters of the squares. (1) Does it look like the table is showing a constant of proportionality for the relationship between the diagonals and the perimeters? In other words, are there the same number of centimeters in the perimeter for every 1 centimeter in the diagonal? (1) Square Diagonal Perimeter Perimeter ÷ Diagonal ------------ A 3 8.4 2.8 B 4.5 12.8 2.8 C 5.7 16 2.8 D 7.6 21 More 2.8
Transcript text: Determine if there is a constant of proportionality for the lengths of the diagonals and the perimeters of the squares. ${ }^{(1)}$ Does it look like the table is showing a constant of proportionality for the relationship between the diagonals and the perimeters? In other words, are there the same number of centimeters in the perimeter for every 1 centimeter in the diagonal? (1) \begin{tabular}{|c|c|c|c|} \hline Square & Diagonal & Perimeter & Perimeter $\div$ Diagonal \\ \hline A & 3 & 8.4 & 2.8 \\ \hline B & 4.5 & 12.8 & 2.8 \\ \hline C & 5.7 & 16 & 2.8 \\ \hline D & 7.6 & 21 More $\downarrow$ & 2.8 \\ \hline \end{tabular}
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Solution

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Solution Steps

Step 1: Identify the relationship between diagonal and perimeter

For a square, the perimeter \( P \) is related to the diagonal \( d \) by the formula: \[ P = d \cdot 2\sqrt{2} \] This is because the side length \( s \) of a square is related to the diagonal by \( s = \frac{d}{\sqrt{2}} \), and the perimeter is \( P = 4s \).

Step 2: Calculate the theoretical constant of proportionality

Using the formula \( P = d \cdot 2\sqrt{2} \), the constant of proportionality is: \[ 2\sqrt{2} \approx 2.828 \]

Step 3: Compare the theoretical constant with the table values

The table shows that for each square, the ratio \( \frac{\text{Perimeter}}{\text{Diagonal}} \) is approximately \( 2.8 \), which matches the theoretical constant \( 2\sqrt{2} \approx 2.828 \). This indicates a constant of proportionality exists between the diagonals and perimeters of the squares.

Final Answer

Yes, there is a constant of proportionality for the lengths of the diagonals and the perimeters of the squares. The constant is approximately \( \boxed{2.828} \).

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