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What Makes Two Shapes Similar Figures?

Two figures are considered to be "similar figures" if they have the same shape, congruent corresponding angles (meaning the angles in the same places of each shape are the same) and equal scale factors. Equal scale factors mean that the lengths of their corresponding sides have a matching ratio. Knowing how to identify similar figures makes it easier to prove geometric theorems and postulates.

There's a difference between similar and congruent figures. Two shapes are congruent when they are the same exact size and have the same angle measurements. Similar figures, on the other hand, do not have to be the same size.

Below is an example of similar shapes:

similar figures: Diagram showing two similar shapes

Although they are different sizes, triangle ABC and triangle DEF are considered similar triangles because they have proportional shapes and angles. Triangle ABC is simply an enlargement of DEF.

Determining Scale Factors of Similar Figures

Below is an example of two similar shape polygons where the ratios of the corresponding sides are equivalent. This means that when you divide each set of corresponding side lengths, you will get the same number. This number is called the scale factor and it can be used to find missing side lengths of a figure.

similar figures: Diagram showing two similar shape polygons where the ratios of the corresponding sides are equivalent

Since polygons ABCD and EFGH are the same shape, we can divide the known side lengths to determine the scale factor. Each corresponding side length will be marked with the same amount of lines. Let’s divide side BD, which is 14 units long, by FH, which is seven units. We’ll then divide side AC, 18 units, by EG, nine units:

similar figures: Scale factor formula

similar figures: Scale factor formula

We now know that the scale factor for these similar figures is 2. From this, we can determine the unknown length of AB. Since side EF corresponds with side AB, we will multiply EF’s measurement of 15 by the scale factor.

similar figures: Formula to determine the length of AB

similar figures: Formula to determine the length of AB

Formula to determine the length of AB

Formula to determine the length of AB

Side CD corresponds with side GH. Because of this, we can use GH’s measurement of 12 to determine the length of CD.

Formula to determine the length of CD

Formula to determine the length of CD

Formula to determine the length of CD

Formula to determine the length of CD

The Importance of Similar Figures

Analyzing similar figures is a concept that’s applicable to many different types of math. Knowing how to identify similar figures and determine their scale factor will make concepts like geometric proofs easier to understand.

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