Lesson 4

Dilating Lines and Angles

4.1: Angle Articulation (10 minutes)

Warm-up

In middle school, students studied many examples of dilations and verified experimentally that dilations preserve angle measure. In this activity, students confirm this claim for one example, and recall what they know about dilations from middle school.

Student Facing

Triangle $$A’B’C’$$ is a dilation of triangle $$ABC$$ using center $$P$$ and scale factor 2.

1. What do you think is true about the angles in $$A’B’C’$$ compared to the angles in $$ABC$$?
2. Use the tools available to figure out if what you thought was true is definitely true for these triangles.
3. Do you think it would be true for angles in any dilation?

Student Response

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Anticipated Misconceptions

If students struggle to come up with a conjecture ask, "What stays the same? What changes?"

Activity Synthesis

Invite students to share their conjectures.

Once students have shared several conjectures, ask students to generalize a claim about angles under dilation. (Dilations keep the measures of angles constant.) They may recall that dilations preserve angle measure from middle school. They may also connect the preservation of angle to the idea that dilations don’t distort the shape of a figure, it just changes its size.

Tell students that we aren’t going to prove that dilations preserve angle measure in this class, we’re going to assert that it is true.

Ask students to add this assertion to their reference charts as you add it to the class reference chart:

If a figure is dilated, then corresponding angles are congruent. (Assertion)

4.2: Dilating Lines (10 minutes)

Activity

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.

Launch

For the sake of time invite students to estimate rather than measure precisely.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to one scale factor greater than one and one scale factor smaller than one.
Supports accessibility for: Organization; Attention

Student Facing

1. Dilate point $$A$$ using center $$C$$ and scale factor $$\frac{3}{4}$$.
2. Dilate point $$B$$ using center $$C$$ and scale factor $$\frac13$$.
3. Dilate point $$D$$ using center $$C$$ and scale factor $$\frac32$$.
4. Dilate line $$CE$$ using center $$C$$ and scale factor 2.
5. What happens when the center of dilation is on a line and then you dilate the line?

Student Response

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Student Facing

• $$X$$ is the midpoint of $$AB$$.
• $$B'$$ is the image of $$B$$ after being dilated by a scale factor of 0.5 using center $$C$$.
• $$A'$$ is the image of $$A$$ after being dilated by a scale factor of 0.5 using center $$C$$.

Call the intersection of $$CX$$ and $$A'B'$$ point $$X'$$. Is point $$X'$$ a dilation of point $$X$$? Explain or show your reasoning.

Student Response

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Anticipated Misconceptions

If students are distracted by all the other points on the diagram, suggest 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 synthesis.

Activity Synthesis

The goal of the synthesis is for students to connect 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 lines 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 the next synthesis, 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.

Speaking: MLR8 Discussion Supports. As students share what happens to the line itself under dilation, press for details by asking how they know that a dilation leaves a line passing through the center of dilation unchanged. Show concepts multi-modally by drawing and labeling a dilation of a line that passes through the center of dilation. Also show students how the dilation takes points on the line to points on that same line. This will help students justify why a dilation does not change a line that passes through the center of dilation.
Design Principle(s): Support sense-making; Optimize output (for justification)

4.3: Proof in Parallel (15 minutes)

Activity

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.

Launch

In the previous lesson you collected many dilations of triangles from students, drawn on tracing paper. Display several of these examples for all to see. Superimpose the examples so that the center of dilation and original figure are lined up. Ask students what they notice about the angles in the problem. (Corresponding angles in the image and original figure are congruent.) Ask students what they notice about the lines in the problem. (They are longer or shorter according to the scale factor. They are parallel.)

If students don’t mention that the lines are parallel, use a highlighter or colored pencil to draw their attention to two corresponding line segments on lines that don’t go through the center. Ask them what they notice about those lines specifically.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof that a dilation takes a line not passing through the center of dilation to a parallel line. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the center and scale factor of the dilation?”, “How do you know that points _____ and _____ will both be on ray _____?” and “How do you know that lines _____ and _____ are parallel?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why a dilation takes lines to parallel lines.
Design Principle(s): Optimize output (for justification); Cultivate conversation

Student Facing

Jada dilated triangle $$ABC$$ using center $$P$$ and scale factor 2.

1. Jada claims that all the segments in $$ABC$$ are parallel to the corresponding segments in $$A’B’C’$$. Write Jada's claim as a conjecture.
3. In Jada’s diagram the scale factor was greater than one. Would your proof have to change if the scale factor was less than one?

Student Response

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Anticipated Misconceptions

For the proof it might be easier to look at one pair of corresponding segments rather than the whole triangle. Recommend 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 segment of interest).

Activity Synthesis

The goal of this synthesis is to conclude 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 share what helped them get started on the proof. (Drawing auxiliary lines. Looking at the reference chart.) Ask students to contribute ideas to the proof until everyone understands this chain of reasoning:

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

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

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)

Lesson Synthesis

Lesson Synthesis

Invite students to sketch each diagram.

“Point $$C$$ was dilated using center $$M$$ by a scale factor of $$\frac25$$. What must be true about $$C$$ and $$C’$$?” ($$C$$ and $$C’$$ are on the same line through $$M, \frac{C’M}{CM}=\frac25$$.)

• If students don’t mention collinearity, sketch for all to see a picture where $$C’$$ is closer to $$M$$ than $$C$$ but clearly not collinear and ask, could this be an accurate picture of $$C, C’$$, and $$M$$?
• If students don’t mention length, sketch for all to see a picture where $$C’$$ is farther from $$M$$ than $$C$$ and ask could this be an accurate picture of $$C, C’,$$ and $$M$$?

“Segment $$NO$$ was dilated using center $$P$$ by a scale factor of $$\frac43$$. What must be true about segments $$NO$$ and $$N’O’$$?” ($$NO$$ and $$N’O’$$ are parallel or on the same line, the ratio of the length of $$N’O’$$ to the length of $$NO$$ is $$\frac43$$).

Ask students to share diagrams of $$NO, P$$, and $$N’O’$$ with a partner and then the class.

“Angle $$CDE$$ was dilated using center $$J$$ by a scale factor of 1.5. What must be true about angle $$C’D’E’$$ compared to angle $$CDE$$?” (They are congruent.)

If students say that angle $$C’D’E’$$ is “bigger” than angle $$CDE$$, ask students to say more about what they mean. Clarify that the segments are longer and points are farther apart but the angles have the same measure.

4.4: Cool-down - All Together Now (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

When one figure is a dilation of the other, we know that corresponding side lengths of the original figure and dilated image are in the same proportion, and are all related by the same scale factor, $$k$$. What is the relationship of corresponding angles in the original figure and dilated image?

For example, if triangle $$ABC$$ is dilated using center $$P$$ with scale factor 2, we can verify experimentally that each angle in triangle $$ABC$$ is congruent to its corresponding angle in triangle $$A’B’C’$$. $$\angle A \cong \angle A’, \angle B \cong \angle B’, \angle C \cong \angle C’$$.

What is the image of a line not passing through the center of dilation? For example, what will be the image of line $$BC$$ when it is dilated with center $$P$$ and scale factor 2? We can use congruent corresponding angles to show that line $$BC$$ is taken to parallel line $$B’C'$$.

What is the image of a line passing through the center of dilation?

For example, what will be the image of line $$GH$$ when it is dilated with center $$C$$ and scale factor $$\frac12$$? When line $$GH$$ is dilated with center $$C$$ and scale factor $$\frac12$$, line $$GH$$ is unchanged, because dilations take points on a line through the center of a dilation to points on the same line, by definition.

So, 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.