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

In the trapezoids above, notice 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:**

- Carefully describe the sequence of similarity transformations necessary to carry the first figure to the second. AND/OR
- 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

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

ADEH=42=2 DCHG=84=2 CBGF=42=2 BAFE=84=2

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

#### Finding the Length of Sides

The two triangles below are similar with

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

∠A corresponds to∠D . Sincem∠D=37∘ ,m∠A=37∘ .BC¯¯¯¯¯¯¯¯ corresponds toEF¯¯¯¯¯¯¯¯ .EFBC=62=3 , so the scale factor is 3.DE¯¯¯¯¯¯¯¯ corresponds toAB¯¯¯¯¯¯¯¯ . This means thatDEAB=3 . You know thatAB=3 , soDE3=3 . This meansDE=9 .

**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

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.

ΔABC∼−−−−− ∠C≅−−−−− FDCA=D−−−

Notice that

\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.

### Review (Answers)

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