Question

diagonal vs. perimeter
here is some of the data that you and your classmates collected
about squares with different diagonal lengths.
diagonal length (toothpicks) perimeter (toothpicks)
1 2.83
2 5.66
3 8.49
4 11.32
5 14.15
a. discuss: is the relationship between diagonal length and
perimeter proportional? how do you know?
b. how can you estimate the perimeter of a square with a
diagonal of 100 toothpicks?

Explanation:

a.

To check for proportionality, we use the formula for proportionality \(y = kx\), where \(k\) is the constant of proportionality. Let \(x\) be the diagonal length and \(y\) be the perimeter.
For \(x = 1,y=2.83\), then \(k=\frac{y}{x}=\frac{2.83}{1}=2.83\)
For \(x = 2,y = 5.66\), then \(k=\frac{y}{x}=\frac{5.66}{2}=2.83\)
For \(x = 3,y=8.49\), then \(k=\frac{y}{x}=\frac{8.49}{3}=2.83\)
For \(x = 4,y = 11.32\), then \(k=\frac{y}{x}=\frac{11.32}{4}=2.83\)
For \(x = 5,y=14.15\), then \(k=\frac{y}{x}=\frac{14.15}{5}=2.83\)

Since the ratio \(\frac{\text{Perimeter}}{\text{Diagonal length}}\) (i.e., \(k\)) is constant (\(k = 2.83\)), the relationship between diagonal length and perimeter is proportional.

b.

Since the relationship is proportional with \(k = 2.83\) (from part a), and the formula for the relationship is \(y=kx\) (where \(y\) is perimeter and \(x\) is diagonal length).
When \(x = 100\) (diagonal length), we substitute \(x\) into the equation \(y=kx\).
So, \(y=2.83\times100\)

Answer:

a. The relationship is proportional. Because \(\frac{\text{Perimeter}}{\text{Diagonal length}}\) is a constant (\(2.83\)).
b. Estimate the perimeter as \(2.83\times100 = 283\) toothpicks.