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9-12 Integrated Math 2 Unit 2 Lesson 5

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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 1 Lesson 2 Lesson 3 Lesson 4 •Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 Lesson 17 Lesson 18

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Lesson 5

5.1 Warm-up 5.2 Activity 5.3 Activity Student Lesson Summary

Unit 2, Lesson 5

Splitting Triangle Sides with Dilation (Part 1)

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Integrated Math 2

5.1

Warm-up

Notice and Wonder: Midpoints

Here’s a triangle with midpoints , and . What do you notice? What do you wonder?

<p>Triangle A B C with midpoints M, L and N drawn forming a triangle.</p>

Are You Ready for More?

5.2

Activity

Dilation or Violation?

Here’s a triangle . Points and are the midpoints of 2 sides.

<p>Triangle A B C. Point M is midpoint of A B. Point N is midpoint of A C. Segment M N is drawn. On both segments A M and M B, one tick mark. On both segments A N and C N, two tick marks. <br>
</p>

Convince yourself that triangle is a dilation of triangle . What is the center of the dilation? What is the scale factor?
Convince your partner that triangle is a dilation of triangle , with the center and scale factor that you found.
With your partner, check the definition of dilation on your reference chart, and make sure both of you could convince a skeptic that definitely fits the definition of dilation.
Convince your partner that segment is twice as long as segment .
Prove that . Convince a skeptic.

Are You Ready for More?

5.3

Activity

A Little Bit Farther Now

Here’s a triangle, . is of the way from to . is of the way from to .

What can you say about segment , compared to segment ? Provide a reason for each of your conjectures.

<p>Triangle A B C with segment M N drawn from A B to A C.</p>

Are You Ready for More?

Student Lesson Summary

Let's examine a segment whose endpoints are the midpoints of 2 sides of the triangle. If is the midpoint of segment and is the midpoint of segment , then what can we say about and triangle ?

Segment is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if , then . This happens because the entire triangle is a dilation of triangle , with a scale factor of .

<p>Triangle A B C. Point E is midpoint of segment B A. Point D is midpoint of segment B C. Segment E D drawn. On both segments A E and E B, two tick marks. On both segments B D and C D, one tick mark.<br>
</p>

In triangle , segment divides segments and proportionally. In other words, =. Again, there is a dilation that takes triangle to triangle , so is parallel to , and we can calculate its length using the same scale factor.

<p>Triangle A B C. Point G on segment A B. Point F on segment B C. Directed line segments A C and G F drawn with arrows point towards Points C and F.<br>
</p>

Glossary

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