8.4.2-Develop Understanding of Dilations and Similarity

Develop

SESSION 2

Understanding of Dilations and Similarity

LESSON 4 • Understand Dilations and Similarity

Lesson Purpose: Develop the idea that a dilation creates an image that is similar to, or the same shape as, the original figure.

Understand how to verify and communicate that two figures are similar.

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A B
C D

Start

Which One Doesn’t Belong?

​Think about equivalent fractions.

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Figure S is dilated to form figure T. The scale
factor is 1.5 and the center of dilation is point A.

a.

How are figures S and T the same?

b.

How are figures S and T different?

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Figures that are the same shape are similar.
Two figures are similar if you can map one
figure onto the other by a sequence of one or
more transformations.

Figures S and T in problem 1 are similar
because you can map figure S onto figure T
using a dilation. You can write the similarity
statement:

figure S ~ figure T

Is figure T similar to figure S? How do
you know?

Match

Match the following angles and sides of $$\Delta ABC$$ to the corresponding angles and sides of $$\Delta DEF$$

• $$\angle A$$

• $$\angle B$$

• $$\angle C$$

• $$\overline{AB}$$

• $$\overline{BC}$$

D

E

F

DE

EF

• D

• E

• F

• DE

• EF

Reorder

Reorder the following to show corresponding angles and sides with pentagon $$QRSTU$$

• E

• F

• G

• H

• I

1

2

3

4

5

• 1
• 2
• 3
• 4
• 5

Poll

What are the corresponding angles?

• $$\angle A\ and\ \ \angle D$$

• $$\angle B\ and\ \angle D$$

• $$\angle A\ and\ \angle F$$

• $$\angle C\ and\ \angle F$$

• $$\angle B\ and\ \angle E$$

Poll

What are the corresponding sides?

• $$\overline{AB}\ and\ \overline{EF}$$

• $$\overline{BC}\ and\ \overline{EF}$$

• $$\overline{BC}\ and\ \overline{DE}$$

• $$\overline{AB}\ and\ \overline{DE}$$

• $$\overline{CA}\ and\ \overline{FD}$$

Multiple Choice

Similar figures have the same ______ but different ______. 

• measure; shapes
• sizes; shapes
• value; measures
• shape; sizes

Multiple Choice

How does a dilation change the angles of a polygon?

• It changes by the scale factor.
• Dilations do not change the angles.
• It is impossible to answer this question.  The angles change unpredictably.
• The angles all change to larger angles.

Multiple Choice

If two figures are similar, the corresponding sides are ______________.

• equal
• congruent
• proportional
• none of these

Multiple Choice

Complete the proportion.

$$\frac{BC}{DM}=\frac{?}{AD}$$

• BC

• AC

• MD

• MC

Multiple Choice

The pair of figures is similar. Find the missing side.

• x = 9
• x = 28
• x = 7
• x = 63

Multiple Choice

Find y.

• 144
• 18
• 72
• 12

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Choose two figures below that appear to
be similar and write a similarity statement
for them.

Multiple Choice

Look at the Rectangles, Which set of figures can you conclude to be similar?

Multiple Choice

Which of the following sets are similar?

Multiple Choice

What is true about the sides of similar figures?

• Corresponding sides are proportional.
• Corresponding sides are congruent.
• Opposite sides are proportional.
• Opposite sides are congruent.

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Ask: How did you decide which two figures
of A, B, C, and D are similar?

Share: I noticed . . . so I decided . . .

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Figure DEFGHI is dilated to form

similar figure D'E'F'G'H'I'. The scale

factor is and the center of dilation is

point X.

a.

Side corresponds to side

The quotient of their lengths can

be written as a fraction:

Write a fraction for each of the

other five pairs of corresponding

side lengths.

Multiple Choice

What side corresponds with AB? 

• BC
• FE
• DE
• FD

Multiple Choice

$$\Delta PQR\ is\ similar\ to\ \Delta STU$$  What side corresponds to side UT?

• PQ

• RQ

• QR

• QP

Multiple Choice

Two objects that are the same shape but not the same size are _______.

• Congruent
• Vertical
• Similar
• Complementary 

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Figure DEFGHI is dilated to form

similar figure D'E'F'G'H'I'. The scale

factor is and the center of dilation is

point X.

b. What is the relationship between

the quotients of corresponding
side lengths? How does this
compare to the scale factor?

c. How do the measures of the

corresponding angles compare?

​What did you notice about the fractions? Explain...





Corresponding angles have _____________________.

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What is the relationship between the
corresponding side lengths and corresponding
angles in similar figures DEFGHI and D'E'F'G'H'I'?

The corresponding side lengths are _________________________ and the corresponding angles are ________________________.

Multiple Choice

All angles in similar figures are congruent.  

• True
• False

Multiple Choice

What is true about the angles of similar figures?

• Corresponding angles are proportional.
• Complementary angles are proportional.
• Corresponding angles are congruent.
• Complementary angles are congruent.

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Ask: Why does the relationship between
the scale factor and the quotients of
corresponding side lengths make sense?

Share: I think the relationship makes
sense because . . .

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Describe two ways you can tell that two
figures are similar.

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Close: Exit Ticket

Label the center of dilation for triangles
XYZ and XY'Z'. Then write a similarity
statement for the triangles.

What should you include in your statement?
Look back at your notes.

What is the symbol for "similarity?"

​Now pick two problems from the "green practice pages" starting on page 89.