Unit 3, Lesson 11

Splitting Triangle Sides with Dilation (Part 2)

Geometry

11.1

Warm-up

Notice and Wonder: Parallel Segments

What do you notice? What do you wonder?

<p>Parallel lines A C and M N slope slightly down from left to right. Point B is below the lines. Segment B A intersects M N at M and A C at A. Segment B C intersects M N at N and AC at C. </p>

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11.2

Activity

Prove It: Parallel Segments

Does a line parallel to one side of a triangle always create similar triangles?

Create several examples. Decide if the conjecture is true or false. If it’s false, make a more specific true conjecture.
Find any additional information that you can be sure is true.
Label it on the diagram.
Write an argument that would convince a skeptic that your conjecture is true.

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11.3

Activity

Preponderance of Proportional Relationships

Find the length of each unlabelled side.

Segments and are parallel.
Triangle D E F with horizontal side D F. Vertex E is below D F. Point A lies on side D E and point B lies on side D F. Line segment A B is drawn and is parallel to side E F. Length of D B is 4, D A is 3, and D E is 7 point 5.
Segments and are parallel. Segment is 12 units long. Segment is 2.5 units long.
Triangle E F G with horizontal side F G. Vertex E is below side F G. Point B lies on side E F and point D lies on side E G. Line segment B D is drawn and is parallel to side F G. Length of F G is 15 and B D is 5.

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Student Lesson Summary

In triangle , segment is parallel to segment . We can show that corresponding angles in triangle and triangle are congruent, so the triangles are similar by the Angle-Angle Triangle Similarity Theorem. There must be a dilation that sends triangle to triangle , and so pairs of corresponding side lengths are in the same proportion. Then we can show that segment divides segments and proportionally. In other words, =.

<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>

For example, suppose is of the way from to and is of the way from to . Then if and , we know that and . What will and equal? Since and , we know that and can show that =.

This argument holds in general. A segment in a triangle that is parallel to one side of the triangle divides the other two sides of the triangle proportionally.

Glossary

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Splitting Triangle Sides with Dilation (Part 2)

Warm-up

Notice and Wonder: Parallel Segments

Are You Ready for More?

Activity

Prove It: Parallel Segments

Are You Ready for More?

Activity

Preponderance of Proportional Relationships

Are You Ready for More?

Student Lesson Summary

Glossary