Question

1. name all similar relationships.

Explanation:

Step1: Calculate the ratios of the sides for each pair of rectangles

For \(R_1\) and \(R_2\):
The ratio of the shorter sides is \(\frac{2}{4}=\frac{1}{2}\), and the ratio of the longer sides is \(\frac{5}{10}=\frac{1}{2}\)
For \(R_1\) and \(R_3\):
The ratio of the shorter sides is \(\frac{1.25}{2}=\frac{5}{8}\), and the ratio of the longer sides is \(\frac{3.125}{5}=\frac{5}{8}\)
For \(R_1\) and \(R_4\):
The ratio of the shorter sides is \(\frac{2}{3}\), and the ratio of the longer sides is \(\frac{5}{7.5}=\frac{2}{3}\)
For \(R_2\) and \(R_3\):
The ratio of the shorter sides is \(\frac{1.25}{4}=\frac{5}{16}\), and the ratio of the longer sides is \(\frac{3.125}{10}=\frac{5}{16}\)
For \(R_2\) and \(R_4\):
The ratio of the shorter sides is \(\frac{4}{3}\), and the ratio of the longer sides is \(\frac{10}{7.5}=\frac{4}{3}\)
For \(R_3\) and \(R_4\):
The ratio of the shorter sides is \(\frac{1.25}{3}=\frac{5}{12}\), and the ratio of the longer sides is \(\frac{3.125}{7.5}=\frac{5}{12}\)

Answer:

All pairs of rectangles (\(R_1 - R_2\), \(R_1 - R_3\), \(R_1 - R_4\), \(R_2 - R_3\), \(R_2 - R_4\), \(R_3 - R_4\)) are similar as their corresponding side ratios are equal.