Lesson 5
Splitting Triangle Sides with Dilation, Part 1
5.1: Notice and Wonder: Midpoints (5 minutes)
Warm-up
The purpose of this warm-up is to elicit conjectures about lines connecting the midpoints of sides in triangles, which will be useful when students formalize and prove these conjectures in later activities in this lesson. While students may notice and wonder many things about these images, lengths, angles, and parallel lines are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that triangles formed by segments connecting midpoints are dilations of the larger triangle, and have several of the properties of dilations.
Launch
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
Student Facing
Here’s a triangle \(ABC\) with midpoints \(L, M\), and \(N\).
What do you notice? What do you wonder?
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information.
5.2: Dilation or Violation? (20 minutes)
Activity
The purpose of this activity is for students to practice using the definitions and properties of dilations in proofs. Students first see 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.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Design Principle(s): Optimize output (for justification); Maximize meta-awareness
Supports accessibility for: Visual-spatial processing
Student Facing
Here’s a triangle \(ABC\). Points \(M\) and \(N\) are the midpoints of 2 sides.
- Convince yourself triangle \(ABC\) is a dilation of triangle \(AMN\). What is the center of the dilation? What is the scale factor?
- Convince your partner that triangle \(ABC\) is a dilation of triangle \(AMN\), with the center and scale factor you found.
- With your partner, check the definition of dilation on your reference chart and make sure both of you could convince a skeptic that \(ABC\) definitely fits the definition of dilation.
- Convince your partner that segment \(BC\) is twice as long as segment \(MN\).
- Prove that \(BC=2MN\). Convince a skeptic.
Student Response
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Anticipated Misconceptions
If students struggle to identify the scale factor without any given side lengths, ask them what they do know (\(M\) is the midpoint) and invite them to make up a few examples with numbers (if \(AM\) is 5 then \(MB\) must be 5 too, so \(AB\) is 10) to see if the scale factor stays the same (yes, it's always 2).
Activity Synthesis
The goal of this activity synthesis is for students to see the value of proving that one figure is a dilation of the other. Invite multiple students to share how they know triangle \(ABC\) is a dilation of triangle \(AMN\). Play the role of skeptic, or invite other students to play the role of skeptic.
If the explanations for why triangle \(ABC\) is a dilation of triangle \(AMN\) 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 \(ABC\) is a dilation of \(AMN\). Record for all to see. (\(BC\) is parallel to \(MN\) or the same line, \(\frac{BC}{MN}=\frac{AC}{AN}\), angle \(ABC\) is congruent to angle \(AMN\).) 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 angles pairs when the lines are cut by a transversal.
- Prove that the lines correspond under a translation or \(180^{\circ}\) rotation.
- Prove that the lines correspond under a dilation.
5.3: A Little Bit Farther Now (10 minutes)
Activity
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:
- write, “it’s a dilation so the line is parallel”
- name the center of the dilation
- name the scale factor of the dilation
- use aspects of the definition of dilation, such as that \(A, M,\) and \(B\) have to be collinear
Launch
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Here’s a triangle \(ABC\). \(M\) is \(\frac23\) of the way from \(A\) to \(B\). \(N\) is \(\frac23\) of the way from \(A\) to \(C\).
What can you say about segment \(MN\), compared to segment \(BC\)? Provide a reason for each of your conjectures.
Student Response
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Student Facing
Are you ready for more?
- Dilate triangle \(DEF\) using a scale factor of -1 and center \(F\).
- How does \(DF\) compare to \(D'F'\)?
- Are \(E\), \(F\), and \(E'\) collinear? Explain or show your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is to support students to make their explanations more rigorous.
Select students who wrote clearly about one aspect of their explanations to share their thinking in this sequence. After each student shares, ask students to check their own conjectures and explanations to see if there are any details they could add based on what they heard.
- A student who wrote, “it’s a dilation so the line is parallel”
- A student who named the center of the dilation
- A student who named the scale factor of the dilation
- A student who used aspects of the definition of dilation, such as that \(A, M,\) and \(B\) have to be collinear
Ask students to add this theorem to their reference charts as you add it to the class reference chart:
If a line divides two sides of a triangle proportionally, the line
must be parallel to the third side of the triangle. (Theorem)
\(\frac12=\frac{1.2}{2.4}\) so \(\overline{AC} \parallel \overline{GF}\)
Lesson Synthesis
Lesson Synthesis
A key concept in this lesson is that the triangles formed by connecting midpoints of sides of the triangle are just a special case of dilating triangles using one vertex as the center, but it has some cool properties, which is why we studied it.
Ask students why it’s useful to prove that a shape is a dilation of another shape. Follow up by asking, “What specifically does it tell you about the shapes if one is a dilation of the other?” (Lines in the dilated figure are parallel to lines in the original figure. Corresponding angles in the two figures are congruent. Corresponding lengths are longer or shorter according to the same ratio given by the scale factor. Distances in the two figures are in the same proportion.)
Ask students what they need to look for to prove that one figure is a dilation of the other using the definition of dilation. Students may reference the fact that the distance from the center to points in the image need to be \(k\) times the distance from the center to corresponding points in the original figure. Ask students if the points could be anywhere as long as the distance is right. Support students to describe how the points in the image need to be on the same ray from the center as the points in the original figure. Invite multiple students to explain why the three small triangles from the image in the warm-up are dilations of the original triangle. Prompt students to refer to both the distances to the center and the rays from the center of the dilation.
5.4: Cool-down - Missing Length (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Let's examine a segment whose endpoints are the midpoints of 2 sides of the triangle. If \(D\) is the midpoint of segment \(BC\) and \(E\) is the midpoint of segment \(BA\), then what can we say about \(ED\) and triangle \(ABC\)?
Segment \(ED\) is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if \(AC=10\), then \(ED=5\). This happens because the entire triangle \(EBD\) is a dilation of triangle \(ABC\) with a scale factor of \(\frac12\).
\(\overline{BD}\cong\overline{DC},\overline{BE}\cong\overline{EA}\)
In triangle \(ABC\), segment \(FG\) divides segments \(AB\) and \(CB\) proportionally. In other words, \(\frac{BG}{GA}\)=\(\frac{BF}{FC}\). Again, there is a dilation that takes triangle \(ABC\) to triangle \(GBF\), so \(FG\) is parallel to \(AC\) and we can calculate its length using the same scale factor.
\(\overleftrightarrow{FG} \parallel \overleftrightarrow{AC}\)
