Question

problems 5 - 7: here is some information about the side lengths of two scaled copies, triangle a and triangle b.

1. complete the table to determine the missing side lengths of each triangle.
2. what is a constant of proportionality in this relationship?
3. what does that constant of proportionality tell you about the triangles?

Explanation:

Step1: Find the constant of proportionality

When side length of triangle \(A = 1\) and side length of triangle \(B=6\), the ratio (constant of proportionality \(k\)) of side length of \(B\) to side length of \(A\) is \(k=\frac{6}{1}=6\).

Step2: Find the missing side - length of triangle \(B\) when side - length of triangle \(A=\frac{1}{2}\)

Using the formula \(y = kx\) (where \(y\) is the side - length of \(B\) and \(x\) is the side - length of \(A\)), substitute \(x = \frac{1}{2}\) and \(k = 6\). Then \(y=6\times\frac{1}{2}=3\).

Step3: Find the missing side - length of triangle \(A\) when side - length of triangle \(B = 8\)

Using the formula \(x=\frac{y}{k}\), substitute \(y = 8\) and \(k = 6\). Then \(x=\frac{8}{6}=\frac{4}{3}\).

Answer:

1. When side - length of \(A=\frac{1}{2}\), side - length of \(B = 3\); when side - length of \(B = 8\), side - length of \(A=\frac{4}{3}\).
2. The constant of proportionality \(k = 6\).
3. The constant of proportionality \(k = 6\) means that each side - length of triangle \(B\) is 6 times the corresponding side - length of triangle \(A\).