Topics
6. Polygons
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United States of AmericaVirginia
Grade 7

6.02 Identify similar figures

Lesson
Practice

Identify similar figures

Similar Figures

Similar figures have corresponding sides that are proportional and corresponding interior angles that are congruent. The symbol \sim is used to represent similarity.

Exploration

The two figures below are similar. Slide the slider to create an enlargement or a reduction of the shape.

Loading interactive...

As you change the scale factor:

  1. Which parts of the shape stay the same?
  2. Which parts of the shape change?
  3. What is special about a scale factor of 1? What name can we use for these shapes?
  4. What do you think a scale factor is?

A scale factor is the ratio between the corresponding sides of two similar figures.

There are two quadrilaterals of the same shape, with dashed rays connecting the corresponding vertices from the smaller shape to the larger shape.

An enlargement is an increase in size without changing the shape. This means a scale factor greater than 1.

There are two quadrilaterals of the same shape, with dashed rays connecting the corresponding vertices from the larger shape to the smaller shape.

A reduction is a decrease in size without changing the shape. This means a scale factor between 0 and 1.

To find the scale factor, we need to figure out what number we can multiply all of the sides of a given figure by to create the similar figure.

Consider these triangles that have the same angle measures but different side lengths.

Equilateral triangles where the first has side lengths of 2 units and the other has side lengths of 6 units.

Notice, each side in the smaller triangle is multiplied by 3 to create the side of the larger triangle. This means that the scale factor is 3.

Consider these polygons. All of their corresponding angles are congruent but we need to check to see whether the corresponding sides are proportional before we decide that the polygons are similar.

The image shows two similar quadrilaterals: A B C D and L M N O. Ask your teacher for more information.

We can write the ratio for each pair of corresponding sides to see if they all have the same ratio.

\overline{LM} corresponds to \overline{AB}, \overline{MN} corresponds to \overline{BC}, \overline{NO} corresponds to \overline{CD}, and \overline{OL} corresponds \overline{DA}.

\dfrac{LM}{AB} = \dfrac{4}{8} = \dfrac{1}{2}

\dfrac{MN}{BC} = \dfrac{7}{14} = \dfrac{1}{2}

\dfrac{NO}{CD} = \dfrac{7}{14} = \dfrac{1}{2}

\dfrac{OL}{DA} = \dfrac{4}{8} = \dfrac{1}{2}

All of the ratios are the same so all of the corresponding sides are proportional. Since we already noticed the corresponding angles are congruent, we can say the figures are similar.

We can write a similarity statement:

LMNO \sim ABCD

The scale factor is \dfrac{1}{2}, making this a reduction in size since the scale factor is between 0 and 1.

Examples

Example 1

Identify the scale factor for each pair of similar figures.

a

LMNO \sim STUV

The image shows two similar rectangles: L M N O and S T U V. Ask your teacher for more information.
Worked Solution
Create a strategy

Use the given similarity statement to identify corresponding sides. Look at how the side lengths of the original figure have changed to create the new figure. What were they all multiplied by?

Apply the idea

Identify the corresponding sides:

\overline{ST} corresponds to \overline{LM}, \overline{TU} corresponds to \overline{MN}, \overline{UV} corresponds to \overline{NO}, and \overline{SV} corresponds to \overline{LO}.

Each side of figure LMNO was multiplied by 3 to create the second figure, STUV.

The scale factor is 3.

Reflect and check

Since the scale factor is larger than one, this is an enlargement.

We could have also set up a ratio of corresponding sides. Since we are told the figures are similar, you only need one pair of corresponding sides to find scale factor.

\displaystyle \dfrac{ST}{LM}\displaystyle =\displaystyle \dfrac{6}{2}Set up a ratio of corresponding sides ST and LM
\displaystyle =\displaystyle 3Divide
b

\triangle LMN \sim \triangle QRS

The image shows two similar triangles: L M N and Q R S. Ask your teacher for more information.
Worked Solution
Create a strategy

Use the given similarity statement or image to identify a pair of corresponding sides. Use the sides to write a ratio and find the scale factor.

Apply the idea

\overline{QR} corresponds to \overline{LM}. Use this to create a ratio:

\displaystyle \dfrac{QR}{LM}\displaystyle =\displaystyle \dfrac{9}{12}Set up a ratio of corresponding sides LM and QR
\displaystyle =\displaystyle \dfrac{3}{4}Simplify the fraction

The scale factor is \dfrac34.

Reflect and check

Since the scale factor is between 0 and 1, this is a reduction.

We could have used any pair of corresponding sides to find scale factor.

\displaystyle \dfrac{RS}{MN}\displaystyle =\displaystyle \dfrac{6}{8}Set up a ratio of corresponding sides MN and RS
\displaystyle =\displaystyle \dfrac{3}{4}Simplify the fraction
\displaystyle \dfrac{SQ}{NL}\displaystyle =\displaystyle \dfrac{3}{4}Set up ratio of corresponding sides MN and RS
\displaystyle =\displaystyle \dfrac{3}{4}Simplify the fraction
c

HJKM \sim NPRS

The image shows two similar parallelograms: HJKM and NPRS. Ask your teacher for more information.
Worked Solution
Create a strategy

Use the given similarity statement or image to identify a pair of corresponding sides. Use the sides to write a ratio and find the scale factor.

Apply the idea

\overline{NP} corresponds to \overline{HJ}. Use this to create a ratio:

\displaystyle \dfrac{NP}{HJ}\displaystyle =\displaystyle \dfrac{17.5}{7}Set up a ratio of corresponding sides LM and QR
\displaystyle =\displaystyle 2.5Divide

The scale factor is 2.5.

Example 2

Determine whether each pair of figures is similar. Justify your reasoning.

a
The image shows two triangles: ABC and DEF. Ask your teacher for more information.
Worked Solution
Create a strategy

Two figures are similar if all corresponding angles are congruent and all corresponding sides are proportional.

Apply the idea

First we will check to see if all of the corresponding angles are congruent.

We can see m \angle{C}=40\degree in \triangle{ABC} and there is no angle in \triangle{DEF} with the same measure.

Since all corresponding interior angles are not congruent, these figures are not similar.

Reflect and check

We did not need to check for proportional sides since the corresponding angles are not congruent. This automatically tells us that these two triangles are not similar.

b
The image shows two rectangles: ABCD and QRST. Ask your teacher for more information.
Worked Solution
Create a strategy

All rectangles have four 90 \degree angles. This means all of the corresponding angles between any two rectangles are always congruent. So we just need to see if all of the corresponding sides are proportional.

Apply the idea

Set up ratios of the corresponding sides:

\displaystyle \dfrac{AB}{QR}\displaystyle =\displaystyle \dfrac{42}{22}Ratio of corresponding sides AB and QR
\displaystyle =\displaystyle \dfrac{21}{11}Simplify the fraction
\displaystyle \dfrac{BC}{RS}\displaystyle =\displaystyle \dfrac{30}{15}Ratio of corresponding sides BC and RS
\displaystyle =\displaystyle 2Simplify the fraction

The ratios are not equal, so the rectangle ABCD is not similar to rectangle QRST.

c
The image shows two kites: FGHJ and XYZW. Ask your teacher for more information.
Worked Solution
Create a strategy

Two figures are similar if all corresponding angles are congruent and all corresponding sides are proportional.

Apply the idea

Using the curved angle markings, we can see that all corresponding interior angles are congruent.

\angle F \cong \angle Z, \angle H \cong \angle X, \angle G \cong \angle Y, and \angle J \cong \angle W

Use the corresponding angles from above part (a) to help identify corresponding sides:

  • \overline{ZY} corresponds to \overline{FG} since \angle F \cong \angle Z and \angle G \cong \angle Y
  • \overline{ZW} corresponds to \overline{FJ} since \angle F \cong \angle Z and \angle J \cong \angle W
  • \overline{WX} corresponds to \overline{JH} since \angle J \cong \angle W and \angle H \cong \angle X
  • \overline{XY} corresponds to \overline{HG} since \angle H \cong \angle X and \angle G \cong \angle Y

Set up ratios of corresponding sides to determine if they are proprotional:

\displaystyle \dfrac{ZY}{FG}\displaystyle =\displaystyle \dfrac{25}{20}Ratio of corresponding sides FG and ZY
\displaystyle =\displaystyle \dfrac{5}{4}Simplify the fraction
\displaystyle \dfrac{ZW}{FJ}\displaystyle =\displaystyle \dfrac{20}{16}Ratio of corresponding sides FJ and ZW
\displaystyle =\displaystyle \dfrac{5}{4}Simplify the fraction
\displaystyle \dfrac{WX}{JH}\displaystyle =\displaystyle \dfrac{15}{12}Ratio of corresponding sides JX and WX
\displaystyle =\displaystyle \dfrac{5}{4}Simplify the fraction
\displaystyle \dfrac{XY}{HG}\displaystyle =\displaystyle \dfrac{30}{24}Ratio of corresponding sides HG and XY
\displaystyle =\displaystyle \dfrac{5}{4}Simplify ratio

All corresponding sides are proportional since they all have the same ratio, or scale factor.

These two figures are similar, since all corresponding angles are congruent and corresponding sides are proportional.

Reflect and check

This represents an enlargement since the scale factor, \dfrac{5}{4}, is greater than 1.

We can write a similarity statement:

FGHJ \sim ZYXW

Idea summary

Figures are similar when both:

  • Corresponding interior angles are congruent, and

  • Corresponding sides are proportional.

Scale factor is the ratio of corresponding sides in similar figures.

Similarity Statements are written as \triangle ABC \sim \triangle XYZ. This can help us identify corresponding parts and set up our ratio of corresponding sides.

\dfrac{XY}{AB} = \dfrac{YZ}{BC} = \dfrac{XZ}{AC}

Outcomes

7.MG.2

The student will solve problems and justify relationships of similarity using proportional reasoning.

7.MG.2c

Given two similar quadrilaterals or triangles, write similarity statements using symbols.

7.MG.2d

Write proportions to express the relationships between the lengths of corresponding sides of similar quadrilaterals and triangles.

7.MG.2e

Recognize and justify if two quadrilaterals or triangles are similar using the ratios of corresponding side lengths.

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