Lesson 6Similarity

Learning Goal

Let’s explore similar figures.

Learning Targets

I can apply a sequence of transformations to one figure to get a similar figure.
I can use a sequence of transformations to explain why two figures are similar.

Lesson Terms

similar

Warm Up: Equivalent Expressions

Use what you know about operations and their properties to write three expressions equivalent to the expression shown.

$10 (2 + 3) - 8 \cdot 3$

Activity 1: Similarity Transformations (Part 1)

Problem 1

Triangle $E G H$ and triangle $L M E$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

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Triangle $E G H$ and triangle $L M E$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

Triangle LME and larger, similar triangle EGH with a line running through their hypontenuse's

Problem 2

Hexagon $A B C D E F$ and hexagon $H G L K J I$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

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Hexagon $A B C D E F$ and hexagon $H G L K J I$ are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.

Hexagon ABCDEF and larger hexagon GHIJKL

Are you ready for more?

The same sequence of transformations takes Triangle $A$ to Triangle $B$ , takes Triangle $B$ to Triangle $C$ , and so on. Describe a sequence of transformations with this property.

A circular grid with a small blue triangle, A near the center and green triangles B-E of varying sizes.

Activity 2: Similarity Transformations (Part 2)

Sketch figures similar to Figure A that use only the transformations listed to show similarity.

A translation and a reflection. Label your sketch Figure B.
Pause here so your teacher can review your work.
A reflection and a dilation with scale factor greater than 1. Label your sketch Figure C.
A rotation and a reflection. Label your sketch Figure D.
A dilation with scale factor less than 1 and a translation. Label your sketch Figure E.

Activity 3: Methods for Translations and Dilations

Your teacher will give you a set of five cards and your partner a different set of five cards. Using only the cards you were given, find at least one way to show that triangle $A B C$ and triangle $D E F$ are similar. Compare your method with your partner’s method. What is the same about your methods? What is different?

A grid with point P in the upper left and then triangle ABC and larger triangle DEF. Side AB is 4 units and BC is 2 units. Side DE is 12 units and EF is 6 units.

Lesson Summary

Let’s show that triangle $A B C$ is similar to triangle $D E F$ :

Two figures are similar if one figure can be transformed into the other by a sequence of translations, rotations, reflections, and dilations. There are many correct sequences of transformations, but we only need to describe one to show that two figures are similar.

Blue triangle ABC and green triangle DEF.

One way to get from $A B C$ to $D E F$ follows these steps:

step 1: reflect across line $f$
step 2: rotate $90^{\circ}$ counterclockwise around $D$
step 3: dilate with center $D$ and scale factor 2

Blue triangle ABC on the left of reflection line f. Reflection green triangle B'C'D on right side of f. Green triangle DB"C" as step 2 inside of larger triangle DEF.

Another way would be to dilate triangle $A B C$ by a scale factor of 2 with center of dilation $A$ , then translate $A$ to $D$ , then reflect over a vertical line through $D$ , and finally rotate it so it matches up with triangle $D E F$ . What steps would you choose to show the two triangles are similar?