Question

1. complete the table to show the side lengths and perimeters for several equilateral triangles.
2. explain why the relationship between the perimeter and side length for an equilateral triangle is proportional.
3. what is a constant of proportionality for the relationship?

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: Find the side length when \(P = 81\)

Given \(P=3s\), then \(s=\frac{P}{3}\). Substituting \(P = 81\), we get \(s=\frac{81}{3}=27\).

Step3: Find the side length when \(P = 315\)

Using \(s=\frac{P}{3}\), substituting \(P = 315\), we have \(s=\frac{315}{3}=105\).

Step4: Explain the proportional relationship

For an equilateral triangle, \(P = 3s\). The ratio \(\frac{P}{s}=\frac{3s}{s}=3\) (constant). So, the relationship is proportional.

Step5: Identify the constant of proportionality

From \(P = 3s\), the constant of proportionality \(k = 3\).

Answer:

• For the table: When perimeter \(P = 81\), side length \(s = 27\); when \(P=315\), side length \(s = 105\).
• The relationship between perimeter \(P\) and side length \(s\) of an equilateral triangle is proportional because \(\frac{P}{s}=3\) (constant).
• The constant of proportionality is \(3\).