standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
The goal of this activity is for students to use the definition of similarity, and to practice applying some of the skills they learned when studying congruent figures to similar figures. Students practice writing a sequence of rigid motions and dilations to take one figure onto another, write accurate similarity statements about the figures, and reason about what must be true about corresponding side lengths.
Launch
Ask students to add this definition to their reference charts as you add it to the class reference chart:
One figure is
similar
to another if there is a sequence of rigid motions and dilations that takes the first figure so that it fits exactly over the second. (Definition)
Translation and dilation takes
\(\triangle ABC\)
onto
\(\triangle FDE\)
so
\(\triangle ABC \sim \triangle FDE\)
[IMG]
Tell students
\(\triangle ABC \sim \triangle FDE\)
is a similarity statement, and we read
\(\sim\)
as “is similar to.”
Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for the sequence of transformations that takes one triangle to the other. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the directed line segment of the translation?” and “What is the center and scale factor of the dilation?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students use precise language to describe the sequence of transformations that takes one triangle to the other.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Representation: Internalize Comprehension.
Activate or supply background knowledge about symbolic notation. Provide students with access to explanations of symbols used in the caption for parallel and perpendicular, and connect to the right angle symbol in the image which shows that two segments are perpendicular.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Are the triangles
similar
?
\(\overline{AB}\parallel\overline{QR},\overline{AB}\perp\overline{AE},\overline{QR}\perp\overline{QT}\)
[IMG]
Write a sequence of transformations (dilation, translation, rotation, reflection) to take one triangle to the other.
Write a similarity statement about the 2 figures, and explain how you know they are similar.
Compare your statement with your partner’s statement. Is there more than one correct way to write a similarity statement? Is there a wrong way to write a similarity statement?
Student Response
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Anticipated Misconceptions
If students struggle to define sequences of rigid motions and dilations, review an example from the card sort.
Activity Synthesis
Ask students to determine if their sequence of rigid motions and dilations will always work for any pair of triangles that have all pairs of corresponding angles congruent and all pairs of corresponding sides lengths in the same proportion. (Probably not, some figures would require rotations and dilations. But the general method of dilating and using rigid motions will always work.)
For students who dilated first, ask:
How do you choose the dilation to use? (Dilate from any vertex, by the scale factor given by the ratio of a pair of corresponding sides.)
Once you dilate, how do you know you’ll be able to use rigid motions to take one triangle to the other? (The triangles will be congruent, because their corresponding angles will remain congruent, and all their side lengths will be multiplied by the value
\(k\)
that you need to make them the same length as the corresponding side. Since all the corresponding angles and all the corresponding sides will match, you’ll be able to take one triangle to the other.)
For students who used rigid motions first, ask:
How do you know rigid motion will line up parts of the figures? (Because corresponding angles are congruent, we’ll be able to line up both of the rays that meet at the vertex by translating and rotating, because you can always take an angle onto another congruent angle.)
How do you know that dilation will then line up the remaining two points? (The sides lengths will be in the same proportion, because we’re given that information, and the angles all line up, so the triangles are dilations of each other.)


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