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Common Core For Geometry

In Lesson 11, students examine the effects of dilating figures from two different centers. By experimental verification, they examine the impact on the two dilations of having two different scale factors, the same two scale factors, and scale factors whose product equals . Each of the parameters of these cases provides information on the centers of the dilations, their scale factors, and the relationship between individual dilations versus the relationship between an initial figure and a composition of dilations.

**Dilations from Different Centers**

Classwork

**Exploratory Challenge 1**

Drawing 2 and Drawing 3 are both scale drawings of Drawing 1

a. Determine the scale factor and center for each scale drawing. Take measurements as needed.

b. Is there a way to map Drawing 2 onto Drawing 3 or map Drawing 3 onto Drawing 2?

c. Generalize the parameters of this example and its results.

**Exercise 1**

Triangle π΄π΅πΆ has been dilated with scale factor 1/2 from centers π1 and π2. What can you say about line segments π΄1π΄2, π΅1π΅2, and πΆ1πΆ2?

**Exploratory Challenge 2**

If Drawing 2 is a scale drawing of Drawing 1 with scale factor π1 and Drawing 3 is a scale drawing of Drawing 2 with scale factor π2, what is the relationship between Drawing 3 and Drawing 1?

a. Determine the scale factor and center for each scale drawing. Take measurements as needed.

b. What is the scale factor going from Drawing 1 to Drawing 3? Take measurements as needed.

c. Compare the centers of dilations of Drawing 1 (to Drawing 2) and of Drawing 2 (to Drawing 3). What do you
notice about these centers relative to the center of the composition of dilations π3?

d. Generalize the parameters of this example and its results.

**Exercise 2**

Triangle π΄π΅πΆ has been dilated with scale factor 2/3 from center π1 to get triangle π΄β²π΅β²πΆβ², and then triangle π΄β²π΅β²πΆβ² is dilated from center π2 with scale factor 1/2 to get triangle π΄β²β²π΅β²β²πΆβ²β². Describe the dilation that maps triangle π΄π΅πΆ to triangle π΄β²β²π΅β²β²πΆβ²β². Find the center and scale factor for that dilation.

**Lesson Summary**

In a series of dilations, the scale factor that maps the original figure onto the final image is the product of all the scale factors in the series of dilations.

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