The purpose of this activity is for students to practice using the definitions and properties of dilations in proofs. Students first describe the dilation by finding its center and scale factor, then convince their partner informally that it’s a dilation, and finally use the definition of dilation to make sure that the triangles really fit all the requirements of being a dilation. Then students convince a partner followed by a skeptic that the triangles have a specific property of dilations. They get an opportunity to write a proof that would convince a skeptic that the triangles really are dilations and, therefore, that they have a property of dilations.

While writing their proof, students must construct a viable argument (MP3).

If students struggle to identify the scale factor without any given side lengths, ask them what they do know ( is the midpoint), and invite them to make up a few examples with numbers (if is 5 then must be 5 too, so is 10) to see if the scale factor stays the same (yes, it's always 2).

The goal of this activity synthesis is for students to see the value of proving that one figure is a dilation of the other.

Direct students’ attention to the reference created using Collect and Display. Ask students to share what they learned about the relationship between segment and segment . Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “dividing the side lengths is always the same” already on it and can be updated with the more precise phrase “corresponding segment lengths have the same ratio.”)

Invite multiple students to share how they know that triangle is a dilation of triangle . Play the role of skeptic, or invite other students to play the role of skeptic. For example, you may ask,

• “How do you know that  is the center?” (Because it does not move after the dilation.)
• “How do you know that the scale factor is 2?”  (Because and are midpoints of the sides, is twice as far from as is, and is twice as far from as is.)

If the explanations for why triangle is a dilation of triangle  are vague, discuss some details of a good proof. Discuss the parts of the definition of dilation that students need to account for in their explanations and some of the reasoning they could use.

Ask students what else they can say for sure must be true, now that they know for sure that is a dilation of . Record for all to see. ( is parallel to or the same line, , angle is congruent to angle .) If students don’t mention parallel lines or congruent angles, ask them to look at their reference chart to see what else they can say must be true.

Emphasize to students that we now have multiple ways to prove that lines are parallel to each other:

• Use corresponding, alternate interior, or other angle pairs when the lines are cut by a transversal.
• Prove that the lines correspond under a translation or rotation.
• Prove that the lines correspond under a dilation.

Students get an additional opportunity to draw conclusions about segments dividing two sides of a triangle proportionally. In this activity, students do not need to formally prove their conjectures. The main focus of this activity is on spotting dilations and recognizing the properties of segments that divide two sides of a triangle proportionally.

Monitor for students who use these approaches, in order of precision:

• Write, “It’s a dilation so the line is parallel”
• Name the center and scale factor of the dilation
• Use aspects of the definition of dilation, such as that and have to be collinear

The purpose of this discussion is to support students to make their explanations more rigorous.

Invite previously selected students to share their conjectures and reasoning. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How does each approach use the information about and dividing the sides of the way down?”
• “How can the dilation be described more precisely?”

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

If a line divides two sides of a triangle proportionally, the line must be parallel to the third side of the triangle. (Theorem)