The purpose of this activity is for students to verify experimentally that dilations take lines through the center of dilation to the same line, even though specific points on the line get farther from or closer to the center of dilation according to the same ratio given by the scale factor. Students first dilate points on given lines, and then they are asked to describe what happens to the lines when they are dilated.

If students are distracted by all the other points on the diagram, suggest that they use tracing paper to trace only the relevant points. Repeat for each question. Then transfer all the points back onto the original diagram before the Activity Synthesis.

The goal of the synthesis is for students to understand the connection between the process of dilating points on a line and dilating the line itself. Specifically, what happens if the line goes through the center of the dilation?

Invite students to share how the definition of dilation can help them answer these questions. Students should have the opportunity to hear and articulate that because dilations, by definition, take points along rays through the center, then dilating a line through the center will take all the points to points on that same line, so the line doesn’t move. It may be hard for students to put into words that the points are dilated, but due to the nature of infinity, the line is not changed, so invite several students to put their explanation into their own words. In a later activity, students will state and record a theorem about lines that do and do not pass through the center of the dilation, so it’s useful for students to be clear about why this is true.

In this activity, students figure out that because dilations preserve angle measures, we can prove that dilations take lines to parallel lines. They draw on the many proofs they did in previous units that use congruent angles to prove that lines are parallel. For students who struggled with proofs in prior units, make sure that they have their reference charts and proof-writing sentence frames available. 

For the proof it might be easier to look at one pair of corresponding segments rather than the whole triangle. Recommend that students look at their reference chart and proof-writing template.

If students are stuck on the proof, encourage them to draw the rays that show how the points in the image were dilated, and to focus on just one pair of corresponding segments at a time (perhaps using colored pencils to highlight the segments of interest).

The goal of this synthesis is to conclude that if two figures are dilations of one another, then any distinct corresponding lines must be parallel. In a later lesson in this unit, students will need to use this result to prove that lines are parallel. Students will get more opportunities to draw conclusions about lines in dilated figures in the Cool-down and Lesson Synthesis. 

Invite students to contribute ideas to the proof until everyone understands this chain of reasoning:

• Because we know that the figures are dilations, we know that corresponding angles are congruent.
• Because we know that corresponding angles are congruent, then we know that the lines cut by the transversal are parallel.

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

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. (Theorem)