2.2.1: Similarity
- Page ID
- 35698
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Lesson
Let's explore similar figures.
Exercise \(\PageIndex{1}\): Equivalent Expressions
Use what you know about operations and their properties to write three expressions equivalent to the expression shown.
\(10(2+3)-8\cdot 3\)
Exercise \(\PageIndex{2}\): Similarity Transformations (Part 1)
1. Triangle \(EGH\) and triangle \(LME\) are similar. Find a sequence of tanslations, rotations, reflections, and dilations that shows this.
2. Hexagon \(ABCDEF\) and hexagon \(HGLKJI\) are similar. Find a sequence of translations, rotations, reflections, and dilations that shows this.
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.
Exercise \(\PageIndex{3}\): 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.
Exercise \(\PageIndex{4}\): 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 \(ABC\) and triangle \(DEF\) are similar. Compare your method with your partner’s method. What is the same about your methods? What is different?
Summary
Let’s show that triangle \(ABC\) is similar to triangle \(DEF\):
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.
One way to get from \(ABC\) to \(DEF\) 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
Another way would be to dilate triangle \(ABC\) 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 \(DEF\). What steps would you choose to show the two triangles are similar?
Glossary Entries
Definition: Similar
Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.
In this figure, triangle \(ABC\) is similar to triangle \(DEF\).
If \(ABC\) is rotated around point \(B\) and then dilated with center point \(O\), then it will fit exactly over \(DEF\). This means that they are similar.
Practice
Exercise \(\PageIndex{5}\)
Each diagram has a pair of figures, one larger than the other. For each pair, show that the two figures are similar by identifying a sequence of translations, rotations, reflections, and dilations that takes the smaller figure to the larger one.
Exercise \(\PageIndex{6}\)
Here are two similar polygons.
Measure the side lengths and angles of each polygon. What do you notice?
Exercise \(\PageIndex{7}\)
Each figure shows a pair of similar triangles, one contained in the other. For each pair, describe a point and a scale factor to use for a dilation moving the larger triangle to the smaller one. Use a measurement tool to find the scale factor.
