Question

problem 5
an equilateral triangle is a triangle where all three sides have the same length.
the table from the previous screen is shown.
explain why the relationship between the perimeter and side length for an equilateral triangle is proportional.

Explanation:

Step1: Recall the formula for the perimeter of an equilateral triangle

The perimeter \(P\) of an equilateral triangle with side length \(s\) is \(P = 3s\).

Step2: Check the ratio of perimeter to side length

For \(s = 3\), \(\frac{P}{s}=\frac{9}{3}=3\).
For \(s = 10\), \(\frac{P}{s}=\frac{30}{10}=3\).
For \(s = 27\), \(\frac{P}{s}=\frac{81}{27}=3\).
For \(s = 105\), \(\frac{P}{s}=\frac{315}{105}=3\).
Since the ratio \(\frac{P}{s}\) is a constant (\(k = 3\)) for all values of \(s\) and \(P\) in the table, the relationship is proportional.

Answer:

The relationship between the perimeter \(P\) and side length \(s\) of an equilateral triangle is proportional because the formula \(P = 3s\) shows that the ratio \(\frac{P}{s}=3\) (a constant) for all pairs of values in the table.