If you think back to our discussion of similar triangles in an earlier lesson, you may notice that we have already talked about ratios. The sides of similar shapes can be compared using ratios. To be similar, the sides of two shapes must have a common ratio—their sides must be comparable in the same way.
Study the examples of similar shapes with common ratios on the tabs below, and answer the question on each tab.
Example 1
Example 2
Since these two rectangles are similar, the sides have a common ratio. To show the width of the smaller rectangle compared to the larger rectangle, you can simply write 2:4 or \(\mathsf{ \frac{2}{4} }\).
To describe how the smaller rectangle's length compares to the larger rectangle, you would write 4:8 or \(\mathsf{ \frac{4}{8} }\).
Notice that both of these ratios can be reduced to a common ratio. What is that ratio?
1:2 or \(\mathsf{ \frac{1}{2} }\)
These two triangles are similar, so the sides have a common ratio. You can write the following three ratios to describe the sides of the smaller triangle compared to the larger triangle.
4:40 or \(\mathsf{ \frac{4}{40} }\)
5:50 or \(\mathsf{ \frac{5}{50} }\)
6:60 or \(\mathsf{ \frac{6}{60} }\)
What common ratio do all of the sides share?
1:10 or \(\mathsf{ \frac{1}{10} }\)
Question
Think about the two examples you just read. What procedure could you follow to identify a common ratio for the sides of similar figures?
Then, decide if you are writing the ratio of smaller to larger or vice versa.
Next, identify the side lengths and then write a ratio that compares one side to the other.
Finally, reduce the ratios or fractions to arrive at the common ratio.
