Unit 3, Lesson 11
Splitting Triangle Sides with Dilation (Part 2)
Geometry
11.1
Warm-up
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
Does a line parallel to one side of a triangle always create similar triangles?
Find any additional information that you can be sure is true.
Label it on the diagram.
Find the length of each unlabelled side.
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.
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.
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, =.
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.