# Similarity

Videos and solutions to help Grade 8 students learn why dilation alone is not enough to determine similarity.

## New York State Common Core Math Module 3, Grade 8, Lesson 8

### Lesson 8 Student Outcomes

• Students know the definition of similar and why dilation alone is not enough to determine similarity.
• Given two similar figures, students describe the sequence of a dilation and a congruence that would map one figure onto the other.

### Lesson 8 Summary

• Similarity is defined as mapping one figure onto another as a sequence of a dilation followed by a congruence (a sequence of rigid motions).
The notation, △ ABC ∼ △ A'B'C' means that △ ABC is similar to △ A'B'C'.

### NYS Math Module 3 Grade 8 Lesson 8

Classwork
Concept Development
A dilation alone is not enough to state that two figures are similar. Consider the following pair of figures:

Example 1
In the picture below we have a triangle ABC, that has been dilated from center 0, by a scale factor of r = 1/2. It is noted by A'B'C'. We also have triangle A''B''C'', which is congruent to triangle A'B'C' (i.e. △ A'B'C' ≅ △ A''B''C'').

Example 2
In the picture below, we have a triangle DEF, that has been dilated from center O, by scale factor r = 3. It is noted by D'E'F'. We also have triangle D''E''F'', which is congruent to triangle D'E'F' (i.e. △ D'E'F' ≅ △ D''E'F'').

Example 3
In the diagram below △ ABC is similar to △ A'B'C'. Describe the sequence of the dilation followed by a congruence that would prove these figures to be similar.

Example 4
In the diagram below, we have two similar figures. Using the notation, we have △ ABC is similar to △ DEF. We want to describe the sequence of the dilation followed by a congruence that would prove these figures to be similar.

Example 5
Knowing that a sequence of a dilation followed by a congruence defines similarity also helps determine if two figures are, in fact, similar. For example, would a dilation map triangle onto triangle ? That is, is △ ABC ∼ △ DEF?

Example 6
Again, knowing that a dilation followed by a congruence defines similarity also helps determine if two figures are, in fact, similar. For example, would a dilation map Figure A onto Figure A'? That is, is Figure A ∼ Figure A'

Exercises 1–4
1. Triangle ABC was dilated from center O by scale factor r = 1/2. The dilated triangle is noted by A'B'C'. Another triangle A''B''C'' is congruent to triangle A'B'C' (i.e. △ A'B'C' ≅ △ A''B''C''). Describe the dilation followed by the basic rigid motion that would map triangle A''B''C'' onto triangle ABC.

2. Describe the sequence that would show △ ABC ∼ △ A'B'C'.

3. Are the two triangles shown below similar? If so, describe the sequence that would prove △ ABC ∼ △ A'B'C'. If not, state how you know they are not similar.

4. Are the two triangles shown below similar? If so, describe the sequence that would prove △ ABC ∼ △ A'B'C'. If not, state how you know they are not similar.