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 ratio for the first pair

For the first pair of side lengths (Triangle A: \(1\) in, Triangle B: \(6\) in), the ratio is \(\frac{6}{1}=6\).

Step2: Use the ratio to find the missing value

Since the ratio is \(6\), for Triangle A with side length \(1\frac{1}{2}=\frac{3}{2}\) in, the side length of Triangle B is \(6\times\frac{3}{2}=9\) in. For Triangle B with side length \(8\) in, the side length of Triangle A is \(\frac{8}{6}=\frac{4}{3}\) in.

Step3: Determine the constant of proportionality

The relationship between the side lengths of Triangle A (\(x\)) and Triangle B (\(y\)) is \(y = 6x\), so the constant of proportionality is \(6\).

Step4: Answer the question about the triangles

Since the side lengths of the two triangles follow the relationship \(y=kx\) (where \(k = 6\)), the triangles are similar (by the Side - Side - Side similarity criterion as the ratios of corresponding sides are equal).

Answer:

1. The completed table:

Side Length of Triangle A (in.)	Side Length of Triangle B (in.)
\(1\frac{1}{2}=\frac{3}{2}\)	\(9\)
\(\frac{4}{3}\)	\(8\)

1. The constant of proportionality is \(6\).

1. The triangles are similar because the ratios of their corresponding side lengths are equal (the constant of proportionality is \(6\)).