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Lesson 14: Similarity NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M2 GEOMETRY Lesson 14: Similarity Student Outcomes . Students understand that similarity is reflexive, symmetric, and transitive. Students recognize that if two triangles are similar, there is a correspondence such that corresponding pairs of angles have the same measure and corresponding sides are proportional. Conversely, they know that if there is a correspondence satisfying these conditions, then there is a similarity transformation taking one triangle to the other respecting the correspondence. Lesson Notes In Lesson 14, students delve more deeply into what it means for figures to be similar. Examples address the properties of similarity and also focus on circles and, finally, triangles. Lessons 12–13 were intentionally not focused on triangles to emphasize that instructors need to move away from equating similarity with triangles or strictly rectilinear figures and rather to think of and teach similarity as a broader concept. With this idea covered in Lessons 12 and 13, attention is shifted to triangles in the second half of this lesson in order to prepare students for triangle similarity criteria in Lessons 15 and 17. Classwork Opening (4 minutes) . As initially mentioned in Lesson 12, two figures in the plane are similar if there is a similarity transformation that takes one to the other. If 퐴 and 퐵 are similar, we write 퐴~퐵 where “~” denotes similarity. Which of the following properties do you believe to be true? Vote yes by raising your hand. For a Figure 퐴 in the plane, do you believe that 퐴~퐴? After taking the vote, show students a simple figure such as the following triangle, and ask them to reconsider if the figure is similar to itself. Allow students 30 seconds to justify their responses in a sentence, and then have them compare reasons with a neighbor MP.3 before sharing out and moving on to the next property. Scaffolding: For struggling students, place the triangles in the coordinate plane. For two figures 퐴 and 퐵 in the plane, do you believe that if 퐴~퐵, then 퐵~퐴? Lesson 14: Similarity 217 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M2 GEOMETRY After taking the vote, show Figures 퐴 and 퐵, and ask them to reconsider the notion that if 퐴 is similar to 퐵, then is 퐵 similar to 퐴. Allow students 30 seconds to justify their responses in a sentence, and then have them compare reasons with a neighbor MP.3 before sharing out and moving on to the next property. For Figures 퐴, 퐵, and 퐶 in the plane, do you believe that if 퐴~퐵 and 퐵~퐶, then 퐴~퐶? After taking the vote, show students Figures 퐴, 퐵, and 퐶, and ask them to reconsider if 퐴 is similar to 퐵 and 퐵 is similar to 퐶, then is 퐴 similar to 퐶. Allow students 30 seconds to justify their responses in a sentence, and then have them compare reasons with a neighbor MP.3 before sharing out and moving on to the next property. Announce that the properties are in fact true, and state them: . For each figure 퐴 in the plane, 퐴~퐴. Similarity is reflexive. If 퐴 and 퐵 are figures in the plane so that 퐴~퐵, then 퐵~퐴. Similarity is symmetric. If 퐴, 퐵, and 퐶 are figures in the plane such that 퐴~퐵 and 퐵~퐶, then 퐴~퐶. Similarity is transitive. In Examples 1 and 2, we form informal arguments to prove why the conditions on similarity must be true. Example 1 (4 minutes) Present the question to the class, and then consider employing any discussion strategies commonly used for a brief brainstorming session, whether it is a whole-group share out, a timed talk-and-turn session with a neighbor, or a Quick Write. Allow about two minutes for whichever strategy is selected and the remaining two minutes for sharing out as a whole group, demonstrating (as needed) any students’ suggestions on the board. Lesson 14: Similarity 218 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M2 GEOMETRY Example 1 We said that for a figure 푨 in the plane, it must be true that 푨~푨. Describe why this must be true. Remember, to show that for two figures to be similar, there must be a similarity transformation that maps one to the other. Are there such transformations to show that 퐴 maps to 퐴? Take multiple suggestions of transformations that map 퐴 to 퐴: . There are several different transformations that map 퐴 onto itself such as a rotation of 0° or a rotation of 360°. A reflection of 퐴 across a line and a reflection right back achieves the same result. MP.3 . A translation with a vector of length 0 also maps 퐴 to 퐴. A dilation with scale factor 1 maps 퐴 to 퐴, and any combination of these transformations also maps 퐴 to 퐴. Therefore, 퐴 must be similar to 퐴 because there are many similarity transformations that map 퐴 to 퐴. This condition is labeled as reflexive because every figure is similar to itself. Example 2 (4 minutes) Present the question to the class, and then consider employing any discussion strategies commonly used for a brief brainstorming session, whether it is a whole-group share out, a timed talk-and-turn session with a neighbor, or a Quick Write. Allow about two minutes for whichever strategy is selected and the remaining two minutes for sharing out as a whole group, demonstrating (as needed) any students’ suggestions on the board. Example 2 We said that for figures 푨 and 푩 in the plane so that 푨~푩, then it must be true that 푩~푨. Describe why this must be true. Now that students have completed Example 1, allow them time to discuss Example 2 among themselves. This condition must be true because for any composition of transformations that maps 퐴 to 퐵, there is a composition of transformations that can undo the first composition. For example, if a translation by vector 푋푌⃑⃑⃑⃑⃑ maps 퐴 to 퐵, then the vector 푌푋⃑⃑⃑⃑⃑ will undo the transformation and map 퐵 to 퐴. A counterclockwise rotation of 90° can be undone by a clockwise rotation of 90°. 1 . A dilation by a scale factor of 푟 = 2 can be undone with a dilation by a scale factor of 푟 = . 2 . A reflection across a line can be undone by a reflection back across the same line. Therefore, it must be true that if a figure 퐴 is similar to a figure 퐵 in the plane (i.e., if there is a similarity transformation that maps 퐴 to 퐵), then there must also be a composition of transformations that can undo that similarity transformation and map 퐵 back to 퐴. This condition is labeled as symmetric because of its likeness to the symmetric property of equality where if one number is equal to another number, then they both must have the same value (if 푎 = 푏, then 푏 = 푎). We leave the third condition, that similarity is transitive, for the Problem Set. Lesson 14: Similarity 219 This work is licensed under a This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. This file derived from GEO-M2-TE-1.3.0 -08.2015 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 14 M2 GEOMETRY Example 3 (10 minutes) In Example 3, students must show that any circle is similar to any other circle. Note the term similar is used here, unlike in Lesson 8 where students proved the dilation theorem for circles. Encourage students to first discuss the question with a partner. Example 3 Based on the definition of similar, how would you show that any two circles are similar? Based on their discussions, provide students with the following cases to help them along: Scaffolding: . Consider the different cases you must address in showing that any two circles . For more advanced are similar to each other: learners, have students a. Circles with different centers but radii of equal length discuss what cases must be considered in order to If two circles have different centers but have radii of equal length, then show similarity between MP.3 the circles are congruent, and a translation along a vector that brings circles. one center to the other will map one circle onto the other. Consider assigning cases b. Circles with the same center but radii of different lengths (a), (b), and (c) to small If two circles have the same center, but one circle has radius 푅 and the groups. Share out results other has radius 푅′, then a dilation about the center with a scale factor of after allowing for group 푅′ discussion. Case (a) is 푟 = maps one circle onto the other so that 푟푅 = 푅′. 푅 likely the easiest to c. Circles with different centers and radii of different lengths consider since the circles If two circles have different centers and radii of different lengths, then are congruent by a the composition of a dilation described in (b) and the translation translation and, therefore, also similar.
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