Lesson Explainer: Similar Polygons | Nagwa Lesson Explainer: Similar Polygons | Nagwa

Lesson Explainer: Similar Polygons Mathematics • First Year of Secondary School

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

Let us begin by reviewing the definition of a polygon.

Definition: 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.

Given two polygons, we say that they are similar if they have the same “shape.” For now, we will use this term loosely in order to motivate this concept, and we will establish a more specific and quantifiable characterization of similar polygons below. The size of similar polygons can differ, but their shape must be the same. For instance, consider two equilateral triangles of different size.

We can see that these two triangles have different sizes, but they are both equilateral triangles. This means that they have the same shape, hence they are similar. In fact, we know that all equilateral triangles have the same shape, while the lengths of their sides may differ. We would like to extend this concept of similarity to general polygons.

For general polygons, their shape is determined by the proportions of the sides and the interior angles. This leads to the definition of similarity that can apply to general polygons.

Definition: Similarity of Polygons

Two polygons are similar if the proportions of the corresponding sides are constant and corresponding angles are equal.

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

Here, as both shapes are rectangles, they contain the same angles. However, for them 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 3÷2=1.5 and 7.5÷5=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.

We can further distinguish polygons into simple or self-intersecting polygons, where simple polygons only have one enclosed region. In this explainer, we will only consider simple polygons.

When we are given that two polygons are similar, it tells us a lot of information concerning the lengths of sides of the polygons as well as the measures of the interior angles. In our first example, we will use the fact that the proportions of the corresponding sides are constant to find the missing length of a side in a similar rectangle.

Example 1: Finding the Length of a Rectangle’s Side given Its Corresponding One in a Similar Rectangle

Given that the rectangles shown are similar, what is 𝑥?

Answer

In this example, we are given that the two rectangles are similar. Recall that two polygons are similar if the proportions of the corresponding sides are constant and the corresponding angles are equal. Since all interior angles of a rectangle are right angles, the corresponding angles will not produce any new information. We can use that fact that the proportions of the corresponding sides are constant, but we first need to identify the corresponding sides in the two rectangles.

In both rectangles, we can see that one side is longer than the adjacent one. In the smaller rectangle, this is more evident since we know the exact lengths of the sides. Here, the side with length 29 cm is longer than the one with length 26 cm. In the larger rectangle, we can see from the diagram that the side with length 58 cm is longer than the side with the unknown length. Hence, we can establish the corresponding sides in the two similar rectangles as highlighted in the same color below.

The proportion of the two longer sides is given by 5829=2.cmcm

Because the rectangles are similar, the proportion of the two shorter sides must also equal 2. This leads to the equation 𝑥26=2.cmcm

Multiplying both sides of this equation by 26 cm gives us 𝑥=52.

In the previous example, we found an unknown length in a rectangle using the fact that it was similar to another polygon. While it was not difficult to identify the corresponding sides in the similar rectangles, this task can be difficult to manage if we only rely on the visuals. This leads to a need for mathematical notations for similar polygons, which we will now discuss.

Recall that we can refer to a polygon by first labeling the vertices and then listing the adjacent vertices successively until we return to the initial vertex. For instance, we can label the vertices of the two rectangles below as follows.

The rectangle on the left can be referred to as 𝐴𝐵𝐶𝐷, and the rectangle on the right can be referred to as 𝑋𝑌𝑊𝑍. However, these are not the only possible labels for these rectangles. In fact, we can begin with any vertex in each rectangle and move either clockwise or counterclockwise around each rectangle, although we commonly move in alphabetical order where possible. Other possible labels for the two rectangles, for instance, are 𝐵𝐴𝐷𝐶 and 𝑌𝑋𝑍𝑊 respectively.

We can now define the mathematical notation used to indicate similar polygons using these labels.

Definition: Notation for Similar Polygons

The similarity of a number of polygons is denoted by the symbol . When denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon.

To understand this rule in the notation of similar polygons, consider the two similar rectangles above. To denote that the rectangle 𝐴𝐵𝐶𝐷 is similar to the second rectangle, we need to identify each vertex in the second rectangle that corresponds to the vertices 𝐴, 𝐵, 𝐶, and 𝐷.

To find the corresponding vertex, we can use the following rules:

  • The measurements of the interior (corresponding) angles of similar polygons at the corresponding vertices are equal.
  • The ratio of the lengths of the (corresponding) sides between two pairs of corresponding vertices is constant.

In the case of our rectangles, the criterion of equal interior angles does not help us to identify the corresponding vertices since all interior angles in a rectangle are equal to 90. Instead, we need to use the criterion involving the proportion of sides.

We have noted that 𝑋𝑌𝐴𝐵=𝑍𝑊𝐷𝐶=1.5.

Because the opposite sides of a rectangle have equal lengths, we also know that 𝑌𝑊𝐵𝐶=𝑋𝑍𝐴𝐷=1.5.

We can highlight, using the same color, each of the two sides involved in these ratios.

Since we have labeled the first rectangle 𝐴𝐵𝐶𝐷, the order of the sides of the rectangle following this order is blue, purple, green, and yellow. If we follow the same ordering in the second rectangle, we obtain the label 𝑋𝑌𝑊𝑍. Hence, we can denote this similarity by 𝐴𝐵𝐶𝐷𝑋𝑌𝑊𝑍.

In the next example, we will identify a corresponding angle in a similar polygon using this notation.

Example 2: Understanding the Similarity between Polygons

Consider two similar polygons 𝐴𝐵𝐶𝐷 and 𝑋𝑌𝑍𝐿. Which angle in 𝐴𝐵𝐶𝐷 corresponds to 𝑌?

Answer

In this example, we need to find the corresponding angle in a similar polygon. Recall that, when we label a pair of similar polygons using vertices, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. In particular, corresponding vertices in similar polygons mean that the corresponding interior angles of the polygons at the vertices have equal measures.

We are looking for the interior angle in quadrilateral 𝐴𝐵𝐶𝐷 that corresponds to 𝑌 in quadrilateral 𝑋𝑌𝑍𝐿. Since vertex 𝑌 appears second in 𝑋𝑌𝑍𝐿, the corresponding vertex in 𝐴𝐵𝐶𝐷 must be the second vertex in this label, which is 𝐵.

Hence, the angle in 𝐴𝐵𝐶𝐷 corresponding to 𝑌 is 𝐵.

In the next example, we will identify the corresponding side in a similar polygon from a given notation.

Example 3: Understanding the Similarity between Polygons

Complete the sentence: If 𝐵𝐺𝐶𝐷𝐿𝑌𝑁𝑍, then 𝐵𝐺𝐶𝐷=𝑁𝑍.

Answer

In this example, we are given that quadrilateral 𝐵𝐺𝐶𝐷 is similar to quadrilateral 𝐿𝑌𝑁𝑍, which is indicated by the symbol . Recall that, when denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Hence, we can match the corresponding pairs of vertices from the given notation as follows: 𝐵𝐿,𝐺𝑌,𝐶𝑁,𝐷𝑍.andandandand

We also recall that the proportion of sides between two corresponding vertices in similar polygons are constant. The left-hand side of the given equation contains the fraction 𝐵𝐺𝐶𝐷, which is the ratio of two sides of the quadrilateral 𝐵𝐺𝐶𝐷. If we replace each of the vertices appearing here with the corresponding vertex from the quadrilateral 𝐿𝑌𝑁𝑍, the ratio should remain the same. This means 𝐵𝐺𝐶𝐷=𝐿𝑌𝑁𝑍.

Hence, the missing side length on the numerator of the fraction in the right-hand side of this equation is 𝐿𝑌.

Now that we have become more familiar with the notation used to define similar polygons, let us consider how to show that two polygons are similar.

How To: Proving the Similarity of Polygons

To prove that two polygons are similar, we need to

  • identify the corresponding vertices,
  • show that the corresponding interior angles have equal measures,
  • and show that the corresponding sides have equal ratio of lengths.

In the next example, we will determine whether or not two given polygons are similar.

Example 4: Verifying Whether the Two Given Polygons Are Similar

Is polygon 𝐴𝐵𝐶𝐷 similar to polygon 𝐺𝐹𝐸𝑋?

Answer

In this example, we need to determine whether the given polygons are similar. Recall that two polygons are similar if the proportions of the corresponding sides are constant and the corresponding angles are equal. Let us begin by verifying whether the corresponding angles in the two polygons are equal.

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 𝐺𝐹𝐸𝑋, the properties of a parallelogram inform us that 𝐸𝑋=𝐹𝐺 and 𝐸𝐹=𝑋𝐺. We also know that 𝐹 is supplementary to 𝐺 and, therefore, 𝑚𝐺=70. Also, opposite angles in a parallelogram have equal measures, so 𝑚𝑋=110 and 𝑚𝐸=70.

We can apply a similar method for the parallelogram 𝐴𝐵𝐶𝐷 to find all its interior angles. Adding these angles, we have the following diagram.

Recall that, when denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Since we need to determine whether 𝐴𝐵𝐶𝐷 and 𝐺𝐹𝐸𝑋 are similar, the corresponding vertices can be identified as follows: 𝐴𝐺,𝐵𝐹,𝐶𝐸,𝐷𝑋.andandandand

The corresponding angles in the two parallelograms are the interior angles at the corresponding vertices. We can see from the diagram above that 𝑚𝐴=𝑚𝐺,𝑚𝐵=𝑚𝐹,𝑚𝐶=𝑚𝐸,𝑚𝐷=𝑚𝑋.

This tells us that the corresponding angles of the two polygons have equal measurements. Next, let us consider the proportions of the corresponding sides of the two polygons. We know that the opposite sides of a parallelogram have equal lengths, so we can find the missing lengths from the diagram as follows.

We can now compute the ratio of the corresponding sides to make sure that they are equal: 𝑋𝐸𝐷𝐶=2613=2,𝐸𝐹𝐶𝐵=2311.5=2,𝐹𝐺𝐵𝐴=2613=2,𝐺𝑋𝐴𝐷=2311.5=2.cmcmcmcmcmcmcmcm

We can see that the ratio of the corresponding sides in the two polygons are equal. This tells us that the two polygons are similar.

Hence, the answer is yes.

We know that, when two polygons are similar, the ratio of the lengths of the corresponding sides must be constant. This leads to the following definition.

Definition: Scale Factor in Similar Polygons

Given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides.

In the next example, we will find the scale factor between similar polygons.

Example 5: Finding the Ratio of Enlargement between Two Similar Quadrilaterals given Their Dimensions

Quadrilaterals 𝐴𝐵𝐶𝐷 and 𝑋𝑌𝑍𝐿 are similar. Determine the ratio of enlargement rounded to two decimal places if necessary.

Answer

In this example, we are given that quadrilaterals 𝐴𝐵𝐶𝐷 and 𝑋𝑌𝑍𝐿 are similar. We need to find the ratio of enlargement, which is the ratio of the length of the corresponding sides in the similar polygons. We begin by choosing a direction for the enlargement. It can be easier to choose a scale factor larger than 1 by placing the larger lengths on the numerator of the ratio.

Let us begin by identifying the corresponding sides. Recall that, when denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Since we are given that quadrilaterals 𝐴𝐵𝐶𝐷 and 𝑋𝑌𝑍𝐿 are similar, we can identify the following pairs of corresponding vertices as follows: 𝐴𝑋,𝐵𝑌,𝐶𝑍,𝐷𝐿.andandandand

The corresponding sides are the sides between a pair of adjacent corresponding vertices. In the given diagram, we are given the length of 𝐵𝐶 in the first quadrilateral. Since the corresponding vertices to 𝐵 and 𝐶 in the second quadrilateral are 𝑌 and 𝑍, respectively, the corresponding side to 𝐵𝐶 is 𝑌𝑍.

Hence, the ratio of the lengths 𝐵𝐶 and 𝑌𝑍 will give us the ratio of enlargement. We can see that 𝐵𝐶=16,𝑌𝑍=17.cmcm

Since 𝑌𝑍 is larger, we will put this length on the numerator. Hence, the ratio of enlargement is 𝑌𝑍𝐵𝐶=1716=1.0625.cmcm

Rounding the ratio to two decimal places, we obtain 1.06.

In the previous example, we found the scale factor, or the ratio of enlargement, in similar polygons. The scale factor can be used to find missing lengths in similar polygons, as we will see in the next example.

Example 6: Finding the Side Length and Angle Measure in Similar Quadrilaterals

Given 𝐴𝐵𝐶𝐷𝑍𝑌𝑋𝐿, find 𝑚𝑋𝐿𝑍 and the length of 𝐶𝐷.

Answer

In this example, we are given that quadrilateral 𝐴𝐵𝐶𝐷 is similar to 𝑍𝑌𝑋𝐿. Recall that, when we label a pair of similar polygons using vertices, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon. Since we are given that quadrilaterals 𝐴𝐵𝐶𝐷 and 𝑍𝑌𝑋𝐿 are similar, we can identify the following pairs of corresponding vertices: 𝐴𝑍,𝐵𝑌,𝐶𝑋,𝐷𝐿.andandandand

In particular, the corresponding interior angles of the polygons at these pairs of vertices have equal measures: 𝑚𝐴=𝑚𝑍,𝑚𝐵=𝑚𝑌,𝑚𝐶=𝑚𝑋,𝑚𝐷=𝑚𝐿.

We want to find 𝑚𝑋𝐿𝑍, which is the same as 𝑚𝐿. We recall that the sum of the interior angles of a quadrilateral must equal 360. We know the measures of two angles in this quadrilateral, 𝑍 and 𝑌. If we can find 𝑚𝑋, we can find 𝑚𝐿 by using this property.

Using the similarity of the two quadrilaterals, we can see that 𝑚𝑋=𝑚𝐶, where 𝑚𝐶=85 is given in the diagram. Hence, 𝑚𝑋=85. We can sum all interior angles of the second quadrilateral and set this equal to 360: 𝑚𝐿+105+109+85=360.

Rearranging this equation so that 𝑚𝐿 is the subject, we obtain 𝑚𝐿=36010510985=61.

Let us find the length of 𝐶𝐷. To find this missing length, we recall the scale factor of similarity. Recall that, given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides. Since the corresponding vertices of 𝐴 and 𝐵 are 𝑍 and 𝑌, respectively, we know that 𝑍𝑌 is a corresponding side of 𝐴𝐵. Hence, the scale factor is given by 𝑍𝑌𝐴𝐵=15075=2.cmcm

This means that we can obtain the lengths of any side in the second polygon by multiplying the correspond length in the first polygon by 2. In particular, this leads to 2×𝐶𝐷=246.2.cm

Dividing both sides of the equation by 2, we obtain 𝐶𝐷=246.22=123.1.cmcm

Hence, 𝑚𝑋𝐿𝑍=61,𝐶𝐷=123.1.cm

In our final example, we will consider a real-world problem involving similar polygons.

Example 7: Finding the Height of a Model given the Scale Factor between It and Another Model given Its Dimensions

A college professor was using a projector to give his lectures. A slide whose dimensions are 11 inches wide and 7 inches high was projected into an image that was 5312 inches wide. Find the height of the projected image.

Answer

In this example, we need to find the height of the image that is created using a projector. We know that a projector enlarges an image without changing its shape. This leads to the concept of similar polygons, which are polygons whose shapes are the same. We can assume that the slide and the projected image are similar polygons.

We are given that the slide is a rectangle that is 11 inches wide and 7 inches high. We are also given that the projected image is 5312 inches wide. Let us draw two similar rectangles with this information with vertices labeled.

Using the notations of similar polygons, we can write 𝐴𝐵𝐷𝐶𝑋𝑌𝑍𝑊.

We need to find the height of the projected image, which is 𝑌𝑍. To find this missing length, we recall the scale factor of similarity. Recall that, given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides. Since 𝐶𝐷 and 𝑊𝑍 are corresponding sides, the scale factor is given by 𝑊𝑍𝐶𝐷=5311=10722.inin

This gives us the scale factor of 10722. This means that we can obtain the length of a side in the projected image by multiplying the length of the corresponding side in the slide by 10722. In particular, this leads to 𝑌𝑍=10722×𝐵𝐷=10722×7=74922=34122.ininin

Hence, the height of the projected image is 34122 in.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • Two polygons are similar if the proportions of the corresponding sides are constant and corresponding angles are equal.
  • The similarity of two polygons is denoted by the symbol . When denoting similar polygons, the order of the vertices appearing in the label of a polygon should correspond to the same order of vertices appearing in the label of the similar polygon.
  • To prove that two polygons are similar, we need to
    • identify the corresponding vertices,
    • show that the corresponding interior angles have equal measures,
    • and show that the corresponding sides have equal ratio of lengths.
  • Given two similar polygons, the scale factor of similarity is the constant ratio of the length of the corresponding sides.

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