Similarity Transformations

Figures that can be carried to each other using one or more rigid transformations followed by a dilation.

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Definition of Similarity

What does similarity have to do with transformations?

Similarity

A similarity transformation is one or more rigid transformations (reflection, rotation, translation) followed by a dilation. When a figure is transformed by a similarity transformation, an image is created that is similar to the original figure. In other words, two figures are similar if a similarity transformation will carry the first figure to the second figure.

In the picture below, trapezoid \begin{align*}ABCD\end{align*} has been reflected, then rotated, and then dilated with a scale factor of 2. The first three trapezoids are all congruent. The final trapezoid is similar to each of the first three trapezoids.

In the trapezoids above, notice that \begin{align*}\angle B \cong \angle B^{\prime \prime \prime} \end{align*}. Also notice that \begin{align*}B^{\prime \prime \prime} C^{\prime \prime \prime}=2BC\end{align*}. In general, similarity transformations preserve angles. Side lengths are enlarged or reduced according to the scale factor of the dilation. This means that similar figures will have corresponding angles that are the same measure and corresponding sides that are proportional.

In order to determine if two shapes are similar, you can:

1. Carefully describe the sequence of similarity transformations necessary to carry the first figure to the second. AND/OR
2. Verify that all corresponding pairs of angles are congruent and all corresponding pairs of sides are proportional.

Determining Similarity

1. Are the two rectangles similar? Explain.

One way to determine whether the two rectangles are similar is to see if a similarity transformation would carry rectangle \begin{align*}ABCD\end{align*} to rectangle \begin{align*}EFGH\end{align*}. First rotate rectangle \begin{align*}ABCD \ 90^\circ\end{align*} counterclockwise about the origin to create rectangle \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}. Then dilate rectangle \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*} about the origin with a scale factor of \begin{align*}\frac{1}{2}\end{align*} to create rectangle \begin{align*}EFGH\end{align*}.

A rotation followed by a dilation is a similarity transformation. Therefore, the two rectangles are similar.

2. Give another explanation for why the two rectangles from #1 are similar.

Another way to check if two shapes are similar is to verify that all corresponding angles are congruent and all corresponding sides are proportional. Because both shapes are rectangles, all angle measures are \begin{align*}90^\circ\end{align*}. Therefore, all pairs of corresponding angles are congruent. For the sides:

• \begin{align*}\frac{AD}{EH}=\frac{4}{2}=2\end{align*}
• \begin{align*}\frac{DC}{HG}=\frac{8}{4}=2\end{align*}
• \begin{align*}\frac{CB}{GF}=\frac{4}{2}=2\end{align*}
• \begin{align*}\frac{BA}{FE}=\frac{8}{4}=2\end{align*}

Because all corresponding side lengths are in the same ratio, they are proportional.

All corresponding angles are congruent and all corresponding sides are proportional, so the rectangles are similar.

The symbol for similarity is \begin{align*}\sim\end{align*}\begin{align*}\Delta ABC \sim \Delta DEF\end{align*} means “triangle \begin{align*}ABC\end{align*} is similar to triangle \begin{align*}DEF\end{align*}”. Just like with congruence statements, the order of the letters matters. A corresponds to \begin{align*}D\end{align*}\begin{align*}B\end{align*} corresponds to \begin{align*}E\end{align*} and \begin{align*}C\end{align*} corresponds to \begin{align*}F\end{align*}.

Finding the Length of Sides

The two triangles below are similar with \begin{align*}\Delta ABC \sim \Delta DEF\end{align*}. What is \begin{align*}m \angle A\end{align*}? What is the length of \begin{align*}\overline{DE}\end{align*}?

Because the triangles are similar, corresponding angles are congruent and corresponding sides are proportional.

• \begin{align*}\angle A\end{align*} corresponds to \begin{align*}\angle D\end{align*}. Since \begin{align*}m \angle D=37^\circ\end{align*}, \begin{align*}m \angle A=37^\circ\end{align*}.
• \begin{align*}\overline{BC}\end{align*} corresponds to \begin{align*}\overline{EF} \end{align*}. \begin{align*}\frac{EF}{BC}=\frac{6}{2}=3\end{align*}, so the scale factor is 3. \begin{align*}\overline{DE}\end{align*} corresponds to \begin{align*}\overline{AB}\end{align*}. This means that \begin{align*}\frac{DE}{AB}=3\end{align*}. You know that \begin{align*}AB=3\end{align*}, so \begin{align*}\frac{DE}{3}=3\end{align*}. This means \begin{align*}DE=9\end{align*}.

Examples

Example 1

Earlier, you were asked what does similarity have to do with transformations?

Similarity transformations produce similar figures. You might think of similar figures as “shapes that are the same shape but different sizes”, but similar figures can always be linked to rigid motions and dilations as well. If two figures are similar, you will always be able to perform a sequence of rigid motions followed by a dilation on one to create the other.

Example 2

Are the two triangles similar? Explain.

Yes. Rotate \begin{align*}\Delta ABC \ 180^\circ\end{align*} about point \begin{align*}P\end{align*}. Then, dilate about point \begin{align*}F\end{align*} with a scale factor of 1.5 to create \begin{align*}\Delta DEF\end{align*}.

A rotation followed by a dilation is a similarity transformation. Therefore, the two triangles are similar.

Example 3

Use the triangles from #2 to fill in the blanks.

• \begin{align*}\Delta ABC \sim \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
• \begin{align*}\angle C \cong \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}
• \begin{align*}\frac{FD}{CA}=\underline{\frac{D}{}}\end{align*}

Notice that \begin{align*}A\end{align*} corresponds to \begin{align*}D\end{align*}\begin{align*}B\end{align*} corresponds to \begin{align*}E\end{align*}, and \begin{align*}C\end{align*} corresponds to \begin{align*}F\end{align*}. This means that \begin{align*}\Delta ABC \sim \Delta DEF\end{align*}.

\begin{align*}\angle C \cong \angle F\end{align*}.

\begin{align*}\frac{FD}{CA}=\frac{ED}{BA}\end{align*} because \begin{align*}\overline{ED}\end{align*} and \begin{align*}\overline{BA}\end{align*} are corresponding sides.

Example 4

Suppose that \begin{align*}\Delta DOG \sim \Delta CAT\end{align*} with \begin{align*}\frac{AT}{OG}=\frac{1}{3}\end{align*}, State what you know about the sides and angles of the two triangles.

\begin{align*}\angle D \cong \angle C\end{align*}, \begin{align*}\angle O \cong \angle A\end{align*}, \begin{align*}\angle G \cong \angle T\end{align*}\begin{align*}\Delta DOG\end{align*} is the bigger triangle because \begin{align*}\frac{AT}{OG}=\frac{1}{3}\end{align*}. This means that \begin{align*}DO=3CA\end{align*}, \begin{align*}OG=3AT\end{align*}, and \begin{align*}DG=3CT\end{align*}.

Review

1. If two shapes are similar, must they be congruent? Explain.

2. If two shapes are congruent, must they be similar? Explain.

\begin{align*}\Delta ABC \sim \Delta DEF\end{align*}. Decide if each statement is true or false and explain your answer.

3. \begin{align*}\frac{AB}{DE}=\frac{BC}{EF}\end{align*}

4. \begin{align*}AC \cdot BC=DF \cdot EF\end{align*}

5. \begin{align*}\angle B \cong \angle E\end{align*}

For #6-#9, are the two triangles similar? If so, give a similarity statement and explain how you know. If not, explain.

6.

7.

8.

9.

For #10-#12, \begin{align*}\Delta BAC \sim \Delta DEF\end{align*}. Note that the triangles below are not drawn to scale.

10. If \begin{align*}m \angle A=85^\circ\end{align*} and \begin{align*}m \angle D=50^\circ\end{align*}, what is \begin{align*}m \angle F\end{align*}?

11. If \begin{align*}AB=5\end{align*}, \begin{align*}ED=2\end{align*}, and \begin{align*}FD=3\end{align*}, what is \begin{align*}CB\end{align*}?

12. If \begin{align*}AC=3\end{align*}\begin{align*}DE=1\end{align*}, and \begin{align*}AC \cong AB\end{align*}, what is \begin{align*}EF\end{align*}?

For #13-#16, \begin{align*}ABCDE \sim FGHIJ\end{align*}. Note that the pentagons below are not drawn to scale.

13. \begin{align*}GF=?\end{align*}

14. \begin{align*}JF=?\end{align*}

15. \begin{align*}ED=?\end{align*}

16. \begin{align*}m \angle J=?\end{align*}

17. Explain two ways to determine whether or not two triangles are similar.

To see the Review answers, open this PDF file and look for section 6.2.

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Dilation To reduce or enlarge a figure according to a scale factor is a dilation.
Similarity Transformation A similarity transformation is one or more rigid transformations followed by a dilation.