Unit 3, Lesson 6

Connecting Similarity and Transformations

Geometry

6.1

Warm-up

<p><strong>Two diagonal line segments intersect at point P. Between two line segments, Quadrilaterals G H F E and A D C B, both with 4 right angles. G H F E looks like a rectangle, A D C B looks like a square.</strong></p>

How do you know that is not a dilation of ?

6.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Your teacher will give you a set of cards. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.
Your teacher will assign you one card. Write the sequence of transformations (translation, rotation, reflection, dilation) to take one figure to the other.
What scale factor is used to go from Figure to Figure ? What scale factor is used to go from Figure to Figure ?

6.3

Activity

Standards Alignment

Building On

Addressing

Building Toward

Are the triangles similar?

<p><strong>Right Triangles A B E & Q R T. A B, 8. A E, 15. B E, 17. Angle B A E, right. Q R, 3 point 2. Q T, 6. R T, 6 point 8. Angle R Q T, right. Angles A E B & Q T R, congruent. Angles A B E & Q R T, congruent.</strong></p>

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 Lesson Summary

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. For example, triangle is similar to triangle ().

What is a rotation and a dilation that will take onto ?

<p><strong>Two parallel horizontal line segments. Top, F D, bottom, E G, point H between two segments. Triangles D H F and E H G drawn. D H labeled 2, H F labeled 2 point 8. H G labeled 5 point 6, H E labeled 4.</strong></p>

The triangles are similar because a rotation of using center will take segment onto segment , because rotations take lines through the center of the rotation to themselves. It will also take segment onto segment for the same reason. Then will be on a ray from through , and will be on a ray from through . Because , a dilation by a scale factor of 2 and center will take to fit exactly onto . Therefore, there is a sequence of rigid motions and dilations that takes onto , and .

<p><strong>Adding on from previous image. Triangles D H F, blue, and E H G drawn. Point D prime on H E. Point F prime on H G. Triangle D prime H prime F prime drawn, green.</strong></p>

Because both rigid motions and dilations leave corresponding angles congruent, all pairs of corresponding angles in similar figures are congruent. In this example, angle is congruent to angle , angle is congruent to angle , and angle is congruent to angle .
Because rigid motions keep lengths congruent and dilations scale them at the same proportion, all pairs of corresponding sides in similar figures are in the same proportion. In this example, .

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

Triangle is similar to triangle because a rotation with center followed by a dilation with center takes to .

<p>Triangle ABC, rotated and then dilated.</p>

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Connecting Similarity and Transformations

Warm-up

Activity

Standards Alignment

Building On

Addressing

G-SRT.2

Building Toward

Activity

Standards Alignment

Building On

G-CO.6

Addressing

G-SRT.2

Building Toward

Student Lesson Summary

Standards Alignment

Building On

Addressing

G-SRT.1.b

Building Toward

G-SRT.2