Learn on PengienVision, Mathematics, Grade 8Chapter 6: Congruence and Similarity

Lesson 7: Understand Similar Figures

In this Grade 8 enVision Mathematics lesson from Chapter 6, students learn to identify and define similar figures by applying sequences of transformations — including rotations, reflections, translations, and dilations — to map one two-dimensional figure onto another. Students practice using scale factors and coordinate rules to complete similarity transformations and verify that corresponding angles are congruent and corresponding side lengths are proportional. The lesson builds core geometry vocabulary and reasoning skills needed to determine whether figures like triangles, trapezoids, and quadrilaterals are similar using coordinate plane analysis.

Section 1

Similarity via Sequences of Transformations

Property

We can also define similarity using transformations. Two figures are similar if you can map one exactly onto the other using a sequence that includes a dilation (to match the size) followed by any rigid motions (translation, reflection, or rotation to match the position).

Congruent = Rigid Motions only.
Similar = Dilation + Rigid Motions.

Examples

Mapping $\Delta ABC$ to $\Delta A'B'C'$ :
- Step 1 (Size): First, look at the sizes. If $A'B'C'$ is twice as big, dilate $\Delta ABC$ by a scale factor of $k = 2$ .
- Step 2 (Position): Now that they are the same size, how do we get the intermediate triangle to park in the final spot? Translate it $3$ units Right and $1$ unit Down.
- The sequence is: Dilation ( $k=2$ ), then Translation.

Explanation

This skill is like solving a two-step puzzle. Always fix the SIZE first! Calculate your scale factor $k$ by comparing one pair of sides. Once you perform the mental dilation, your shape is now a congruent "clone" of the target. Your second step is simply deciding whether you need to slide it, flip it, or turn it to make it snap perfectly into place.

Section 2

Describing a Sequence of Similarity Transformations

Property

To describe the sequence of transformations that maps a pre-image to a similar image, first determine the scale factor ( $k$ ) and then identify the rigid transformation.

Find the Scale Factor ( $k$ ): Calculate the ratio of a side length in the image to the corresponding side length in the pre-image.
$k = \frac{{\text{image length}}}{{\text{pre-image length}}}$
Find the Rigid Transformation: Apply the dilation to the pre-image to create an intermediate figure. Then, find the translation, reflection, or rotation that maps the intermediate figure onto the final image.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Similarity via Sequences of Transformations

Property

Congruent = Rigid Motions only.
Similar = Dilation + Rigid Motions.

Examples

Mapping $\Delta ABC$ to $\Delta A'B'C'$ :
- Step 1 (Size): First, look at the sizes. If $A'B'C'$ is twice as big, dilate $\Delta ABC$ by a scale factor of $k = 2$ .
- Step 2 (Position): Now that they are the same size, how do we get the intermediate triangle to park in the final spot? Translate it $3$ units Right and $1$ unit Down.
- The sequence is: Dilation ( $k=2$ ), then Translation.

Explanation

Section 2

Describing a Sequence of Similarity Transformations

Property

To describe the sequence of transformations that maps a pre-image to a similar image, first determine the scale factor ( $k$ ) and then identify the rigid transformation.

Find the Scale Factor ( $k$ ): Calculate the ratio of a side length in the image to the corresponding side length in the pre-image.
$k = \frac{{\text{image length}}}{{\text{pre-image length}}}$
Find the Rigid Transformation: Apply the dilation to the pre-image to create an intermediate figure. Then, find the translation, reflection, or rotation that maps the intermediate figure onto the final image.

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Similarity via Sequences of Transformations

Property

Congruent = Rigid Motions only.
Similar = Dilation + Rigid Motions.

Examples

Mapping $\Delta ABC$ to $\Delta A'B'C'$ :
- Step 1 (Size): First, look at the sizes. If $A'B'C'$ is twice as big, dilate $\Delta ABC$ by a scale factor of $k = 2$ .
- Step 2 (Position): Now that they are the same size, how do we get the intermediate triangle to park in the final spot? Translate it $3$ units Right and $1$ unit Down.
- The sequence is: Dilation ( $k=2$ ), then Translation.

Explanation

Section 2

Describing a Sequence of Similarity Transformations

Property

To describe the sequence of transformations that maps a pre-image to a similar image, first determine the scale factor ( $k$ ) and then identify the rigid transformation.

Find the Scale Factor ( $k$ ): Calculate the ratio of a side length in the image to the corresponding side length in the pre-image.
$k = \frac{{\text{image length}}}{{\text{pre-image length}}}$
Find the Rigid Transformation: Apply the dilation to the pre-image to create an intermediate figure. Then, find the translation, reflection, or rotation that maps the intermediate figure onto the final image.