Dilations from Different Centers

New York State Common Core Math Geometry, Module 2, Lesson 11

Worksheets 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 𝐴𝐵𝐶 has been dilated with scale factor 1/2 from centers 𝑂1 and 𝑂2. What can you say about line segments 𝐴1𝐴2, 𝐵1𝐵2, and 𝐶1𝐶2?

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Exploratory Challenge 2

If Drawing 2 is a scale drawing of Drawing 1 with scale factor 𝑟1 and Drawing 3 is a scale drawing of Drawing 2 with scale factor 𝑟2, 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 𝑂3?
d. Generalize the parameters of this example and its results.

Exercise 2

Triangle 𝐴𝐵𝐶 has been dilated with scale factor 2/3 from center 𝑂1 to get triangle 𝐴′𝐵′𝐶′, and then triangle 𝐴′𝐵′𝐶′ is dilated from center 𝑂2 with scale factor 1/2 to get triangle 𝐴′′𝐵′′𝐶′′. Describe the dilation that maps triangle 𝐴𝐵𝐶 to triangle 𝐴′′𝐵′′𝐶′′. 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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