Explainer: Similar Polygons

In this explainer, we will learn how to use the properties of similar polygons to find unknown angles, side lengths, scale factors, and perimeters.

Before we start looking at similar polygons, we first need to recap two things. What is a polygon? And what is similarity?

Definition: The Definition of a Polygon

A polygon is a closed shape with straight sides.

Examples of shapes that are and are not polygons can be seen in the table.

The definition of similarity is given below.

Definition: The Definition of Mathematical Similarity

Two shapes are similar if they have corresponding sides that are proportional and angles that are equal.

An example of two shapes that are similar are the rectangles shown below:

Here, as both shapes are rectangles, they contain the same angles. However, to be similar, we also need to check that the sides of the two rectangles are in proportion. If we divide the corresponding lengths of the two rectangles, we get 3รท2=1.5 and 7.5รท5=1.5. The scale factor between the two sides is constant, and the two rectangles are, therefore, similar. In fact, the two rectangles in this example are polygons and are, therefore, an example of similar polygons.

Now, let us recall some notation used when studying similar polygons. The vertices of a polygon are often labeled with letters in a clockwise direction and the polygon is often referred to using these letters. For example, the polygon in the picture has its vertices labeled ๐ด, ๐ต, ๐ถ, ๐ท, and ๐ธ, and would be referred to as ๐ด๐ต๐ถ๐ท๐ธ.

If two shapes are similar, for example, the triangles ๐ด๐ต๐ถ and ๐ท๐ธ๐น, then we can say ๐ด๐ต๐ถโˆผ๐ท๐ธ๐น. If we know that two shapes are similar, then we know that their corresponding angles are equal and their corresponding sides are in proportion. The converse, or reverse, is also true: If the corresponding angles of two shapes are equal and their corresponding sides are in proportion, then the two shapes are similar.

We can, therefore, use these two facts to solve problems involving similar polygons. There are generally two types of questions that you could be asked. The first provides you with the information that the two shapes are similar and then asks you to use this property to find unknown information (using properties of similarity). The second tells you some information about the two shapes and asks you to determine whether the two shapes are similar (proving similarity). On discovery of similarity, questions may then ask you to use the properties of similarity to find additional information.

Let us look at an example of the first type of question.

Example 1: Using Properties of Similarity to Solve Problems

Given that the rectangles shown are similar, what is ๐‘ฅ?

Answer

As we are told that the two rectangles are similar, we know that their sides must be in proportion. In other words, there must be a unique scale factor between the corresponding sides. The side of the smaller rectangle with length 21 cm corresponds to the side on the larger rectangle with length ๐‘ฅ cm and the side of the smaller rectangle with length 15 cm corresponds to the side on the larger rectangle with length 60 cm. We can work out the scale factor from the smaller rectangle to the larger rectangle by dividing 60 by 15. If we wanted to work in the opposite direction (from larger to smaller), we would divide 15 by 60 to find the scale factor. Generally, it is easier to work in the direction of smallest to largest, so let us do that. The scale factor is equal to 60รท15=4, which tells us that the bigger rectangle is four times larger than the smaller rectangle. Therefore, to find the length ๐‘ฅ, we multiply 21 by 4. So, ๐‘ฅ=21ร—4=84.

Let us look at another example.

Example 2: Using Properties of Similarity to Solve Problems

If the two following polygons are similar, find the value of ๐‘ฅ.

Answer

Here we have two quadrilaterals that we are told are similar. We need to find the scale factor that maps one to the other. We know that the side on the larger quadrilateral with length 85 cm corresponds to the side with length 34 cm on the smaller quadrilateral. If we calculate the scale factor in the direction of the larger shape to the smaller shape, we get 34รท85.

In this case, the scale factor is not a whole number, so we will leave our answer as a simplified fraction: 25. We, therefore, know that the smaller quadrilateral is 25 of the size of the larger quadrilateral. Hence, to find ๐‘ฅ, we multiply 75 by 25: ๐‘ฅ=75ร—25=(75รท5)ร—2=30.

Now let us look at a question where we need to decide if two polygons are similar. There are two criteria that we need to check:

  1. Are the measures of the corresponding angles in each shape equal?
  2. Are the corresponding sides of each shape in proportion?

We will demonstrate this in an example.

Example 3: Proving Two Polygons Are Similar

Is polygon ๐ด๐ต๐ถ๐ท similar to polygon ๐บ๐น๐ธ๐‘‹?

Answer

The first thing to note here is that the two polygons are parallelograms which allows us to calculate the unknown sides and angle measures of each shape. If we consider ๐บ๐น๐ธ๐‘‹, the properties of a parallelogram inform us that ๐ธ๐‘‹=๐น๐บ and ๐ธ๐น=๐‘‹๐บ. We also know that โˆ ๐น is supplementary to โˆ ๐บ and, therefore, ๐‘šโˆ ๐บ=70โˆ˜. Also, opposite angles in a parallelogram have equal measure, so ๐‘šโˆ ๐‘‹=110โˆ˜ and ๐‘šโˆ ๐ธ=70โˆ˜.

A similar argument can be applied to ๐ด๐ต๐ถ๐ท to show that ๐ด๐ต=๐ท๐ถ, ๐ต๐ถ=๐ด๐ท, ๐‘šโˆ ๐ต=110โˆ˜, ๐‘šโˆ ๐ถ=70โˆ˜, and ๐‘šโˆ ๐ท=110โˆ˜. Hence, the corresponding angles in each polygon have equal measure. To prove similarity, we just need to check that the sides are in proportion. We need to check that ๐ธ๐‘‹๐ถ๐ท=๐ธ๐น๐ถ๐ต: ๐ธ๐‘‹๐ถ๐ท=2613=2,๐ธ๐น๐ถ๐ต=2311.5=2.

The corresponding angle measures are equal and the corresponding sides are in proportion and, hence, the polygons are similar.

To finish, let us look at one final example. This time we are going to be asked to determine if two shapes are similar and then state an additional piece of information about the two polygons.

Example 4: Proving Two Polygons Are Similar

Are these two polygons similar? If yes, find the scale factor from ๐‘‹๐‘Œ๐‘๐ฟ to ๐ด๐ต๐ถ๐ท.

Answer

We can see from the question that three of the corresponding angles in the two polygons have equal measure. We can conclude that the fourth angle must also be equal in both polygons. Hence, the measures of the corresponding angles are equal in both quadrilaterals. We then need to check that the corresponding sides are in proportion. If we look carefully at the diagram and the positions of the angles, we can see that ๐‘๐ฟ corresponds to ๐ถ๐ท, ๐ฟ๐‘‹ to ๐ท๐ด, ๐‘‹๐‘Œ to ๐ด๐ต, and ๐‘Œ๐‘ to ๐ต๐ถ. So, we need to check that ๐‘๐ฟ๐ถ๐ท=๐ฟ๐‘‹๐ท๐ด=๐‘‹๐‘Œ๐ด๐ต=๐‘Œ๐‘๐ต๐ถ: ๐‘๐ฟ๐ถ๐ท=3.22.56=54,๐ฟ๐‘‹๐ท๐ด=3.42.72=54,๐‘‹๐‘Œ๐ด๐ต=4.83.84=54,๐‘Œ๐‘๐ต๐ถ=3.22.56=54.

As the corresponding angles have equal measure and the corresponding sides are in proportion, the two quadrilaterals are similar. The scale factor from ๐‘‹๐‘Œ๐‘๐ฟ to ๐ด๐ต๐ถ๐ท is 45=0.8 as we are defining the direction from the bigger shape to the smaller shape.

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