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Similarity Transformations

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Definition of Similarity

What does similarity have to do with transformations?

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https://www.youtube.com/watch?v=-Z0f_RgnAI4

Guidance

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  ABCD 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 \angle B \cong \angle B^{\prime \prime \prime} . Also notice that B^{\prime \prime \prime} C^{\prime \prime \prime}=2BC . 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.

Example A

Are the two rectangles similar? Explain.

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

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

Example B

Give another explanation for why the two rectangles from Example A are similar.

Solution: 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 90^\circ . Therefore, all pairs of corresponding angles are congruent. For the sides:

  • \frac{AD}{EH}=\frac{4}{2}=2
  • \frac{DC}{HG}=\frac{8}{4}=2
  • \frac{CB}{GF}=\frac{4}{2}=2
  • \frac{BA}{FE}=\frac{8}{4}=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.

Example C

The symbol for similarity is \sim\Delta ABC \sim \Delta DEF means “triangle  ABC is similar to triangle DEF ”. Just like with congruence statements, the order of the letters matters. A corresponds to DB corresponds to  E and  C corresponds to F .

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

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

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

Concept Problem Revisited

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.

Vocabulary

Rigid transformations are transformations that preserve distance and angles. The rigid transformations are reflections, rotations, and translations.

Two figures are congruent if a sequence of rigid transformations will carry one figure to the other.  Congruent figures will always have corresponding angles and sides that are congruent as well.

similarity transformation is one or more rigid transformations followed by a dilation.

dilation is an example of a transformation that moves each point along a ray through the point emanating from a fixed center point P , multiplying the distance from the center point by a common scale factor,   k .

Two figures are  similar if a similarity transformation will carry one figure to the other.  Similar figures will always have corresponding angles congruent and corresponding sides proportional.

Guided Practice

1. Are the two triangles similar? Explain.

2. Use the triangles from #1 to fill in the blanks.

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

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

Answers:

1. Yes. Rotate  \Delta ABC \ 180^\circ about point P . Then, dilate about point  F with a scale factor of 1.5 to create \Delta DEF .

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

2. a. Notice that  A corresponds to DB corresponds to E , and  C corresponds to F . This means that \Delta ABC \sim \Delta DEF .

b. \angle C \cong \angle F .

c.  \frac{FD}{CA}=\frac{ED}{BA} because  \overline{ED} and  \overline{BA} are corresponding sides.

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

Practice

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

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

\Delta ABC \sim \Delta DEF . Decide if each statement is true or false and explain your answer.

3. \frac{AB}{DE}=\frac{BC}{EF}

4. AC \cdot BC=DF \cdot EF

5. \angle B \cong \angle E

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, \Delta BAC \sim \Delta DEF . Note that the triangles below are not drawn to scale.

10. If  m \angle A=85^\circ and m \angle D=50^\circ , what is m \angle F ?

11. If AB=5 , ED=2 , and FD=3 , what is CB ?

12. If  AC=3DE=1 , and AC \cong AB , what is EF ?

For #13-#16, ABCDE \sim FGHIJ . Note that the pentagons below are not drawn to scale.

13.  GF=?

14.  JF=?

15.  ED=?

16.  m \angle J=?

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

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