Question

practice problems
1 mai measured the height, perimeter, and area of some equilateral triangles. her
measurements are shown in the tables.
a. could the relationship between the
triangles heights and their perimeters be
proportional? explain your reasoning.
b. could the relationship between the
triangles heights and their areas be
proportional? explain your reasoning.

Explanation:

a. Check proportionality for height and perimeter

Step1: Recall proportionality condition

For two quantities \(x\) (height) and \(y\) (perimeter) to be proportional, the ratio \(\frac{y}{x}\) must be constant.

Step2: Calculate ratios

• For \(x = 1.1\), \(y=3.9\), \(\frac{y}{x}=\frac{3.9}{1.1}\approx3.55\)
• For \(x = 2.4\), \(y = 8.1\), \(\frac{y}{x}=\frac{8.1}{2.4}=3.375\)
• Since the ratios \(\frac{3.9}{1.1}

eq\frac{8.1}{2.4}\), the relationship is not proportional.

b. Check proportionality for height and area

Step1: Recall proportionality condition

For two quantities \(x\) (height) and \(y\) (area) to be proportional, the ratio \(\frac{y}{x}\) must be constant.

Step2: Calculate ratios

• For \(x = 1.1\), \(y = 0.715\), \(\frac{y}{x}=\frac{0.715}{1.1}=0.65\)
• For \(x = 2.4\), \(y = 3.24\), \(\frac{y}{x}=\frac{3.24}{2.4}=1.35\)
• Since the ratios \(\frac{0.715}{1.1}

eq\frac{3.24}{2.4}\), the relationship is not proportional.

Answer:

a. No. The ratios of perimeter to height are not constant (\(\frac{3.9}{1.1}\approx3.55\) and \(\frac{8.1}{2.4}=3.375\)).
b. No. The ratios of area to height are not constant (\(\frac{0.715}{1.1}=0.65\) and \(\frac{3.24}{2.4}=1.35\)).