Lesson 2 Triangle Dilations Solidify Understanding

Jump Start

Given $△ R S T \sim △ A B C$ , find the missing sides of each triangle.

Learning Focus

Create similar figures by dilation given the scale factor.

Prove a theorem about the midlines of a triangle using dilations.

How do I know if two geometric figures are similar?

How do I know if two geometric figures are similar by dilation?

What interesting characteristics of an image are produced by dilating a polygon centered at one of the vertices of the polygon?

Open Up the Math: Launch, Explore, Discuss

1.

Given $△ A B C$ , use point $M$ as the center of a dilation with a scale factor of $3$ to locate the vertices of a new triangle, $A^{'} B^{'} C^{'}$ .
Now use point $N$ as the center of a dilation with a scale factor of $\frac{1}{2}$ to locate the vertices of a new triangle, $A^{″} B^{″} C^{″}$ .

2.

Label the vertices in the two triangles you created in the diagram above. Based on this diagram, write several proportionality statements you believe are true. First write your proportionality statements using the names of the sides of the triangles in your ratios. Then verify that the proportions are true by replacing the side names with their measurements, measured to the nearest millimeter. (Try to find at least 10 proportionality statements you believe are true.)

My list of proportions:

3.

Based on your work, under what conditions are the corresponding line segments in an image and its pre-image parallel after a dilation? That is, which word best completes this statement:

After a dilation, corresponding line segments in an image and its pre-image are parallel.

A.

never

B.

sometimes

C.

always

4.

Give reasons for your answer. If you choose “sometimes,” be very clear in your explanation how to tell when the corresponding line segments before and after the dilation are parallel and when they are not.

5.

Given $△ A B C$ , use point $A$ as the center of a dilation with a scale factor of $2$ to locate the vertices of a new triangle, $A^{'} B^{'} C^{'}$ .

6.

Explain how the diagram you created above can be used to prove the following theorem:

The segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.

Ready for More?

In problem 1, $△ A B C$ was transformed using the two centers of dilation, points $M$ and $N$ , to create two new triangles, one larger than $△ A B C$ and one that was smaller. Locate a third center of dilation for the small and large triangles formed by the dilations described in the task.

a.

b.

What is the scale factor for this new dilation?

Takeaways

Key relationships between an image and its pre-image after a dilation:

Adding Notation, Vocabulary, and Conventions

A dilation is a transformation of the plane, such that if $O$ is the center of the dilation and a nonzero number $k$ is the scale factor, then $P^{'}$ is the image of point $P$ if

A two-dimensional figure is similar to another if

Vocabulary

midline of a triangle
midline of a triangle theorem
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we extended our understanding of similar figures. Since corresponding segments of similar figures are proportional, and dilations produce similar figures, corresponding parts of an image and its pre-image after a dilation are proportional. We also learned that corresponding line segments in a dilation are parallel. These two observations provided a tool for proving a theorem about the midlines of a triangle, a segment connecting the midpoints of two sides of a triangle.

Retrieval

1.

Alternate Interior
Linear Pair
Vertical Angles
Same Side Exterior
Corresponding
Alternate Exterior
Complementary
Same Side Interior

2.

Describe the transformation(s) that occurred between the pre-image and image.

3.

Describe the transformation(s) that occurred between the pre-image and image.