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9-12 Integrated Math 2 Unit 2 Lesson 4

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K-5 Math 6-8 Math •9-12 Math

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Integrated Math 1 •Integrated Math 2 Integrated Math 3

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Lesson 4

4.1 Warm-up 4.2 Activity 4.3 Activity Student Lesson Summary

Unit 2, Lesson 4

Dilating Lines and Angles

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Integrated Math 2

4.1

Warm-up

Angle Articulation

Triangle is a dilation of triangle using center and a scale factor of 2.

<p>Smaller Triangle A B C and larger triangle A prime B prime C prime with Point P. Point P at lower left corner.</p>

What is the same about the two triangles? What is different? Make a conjecture about what stays the same after dilation.
Use the tools available to figure out if what you thought was true is definitely true for these triangles.
Do you think your conjecture will be true for any figure after dilation?

Are You Ready for More?

4.2

Activity

Dilating Lines

<p>Lines A D and C E, intersect at point C. Both lines move upwards and to the right; Line A D is steeper than line C E. Point B on line A D, between points A and C.<br>
</p>

Dilate point using center and a scale factor of .
Dilate point using center and a scale factor of .
Dilate point using center and a scale factor of .
Dilate line using center and a scale factor of 2.
What happens when the center of dilation is on a line and then you dilate the line?

Are You Ready for More?

4.3

Activity

Proof in Parallel

Jada dilates triangle using center and a scale factor of 2.

<p>Smaller Triangle A B C and larger triangle A prime B prime C prime with Point P. Point P at lower left corner.</p>

Prove the conjecture from your whole-class discussion.
In Jada’s diagram the scale factor is greater than one. Would your proof have to change if the scale factor is less than one?

Are You Ready for More?

Student Lesson Summary

When one figure is a dilation of the other, we know that corresponding side lengths of the original figure and the dilated image are in the same proportion, and are all related by the same scale factor, . What is the relationship of corresponding angles in the original figure and the dilated image?

For example, if triangle is dilated, using center , with a scale factor of 2, we can verify experimentally that each angle in triangle is congruent to its corresponding angle in triangle . is congruent to . is congruent to . is congruent to .

<p>Figure showing triangle A B C dilated to triangle A’ B’ C’ using center P and scale factor 2.</p>

What is the image of a line not passing through the center of dilation? For example, what will be the image of line when it is dilated with center and a scale factor of 2? We can use congruent corresponding angles to show that line is taken to parallel line .

<p>Figure showing triangle A B C dilated to triangle A’ B’ C’ using center P.</p>

What is the image of a line passing through the center of dilation?

<p>Line G H with point C on the right.</p>

For example, what will be the image of line when it is dilated with center and a scale factor of ? When line is dilated with center and a scale factor of , line is unchanged, because dilations take points on a line through the center of a dilation to points on the same line, by definition.

<p>Line G H with point C on the right. Points G prime and H prime between points H and C, closer together than points G and H.<br>
</p>

So a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

Glossary

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