Learn on PengiReveal Math, AcceleratedUnit 6: Congruence and Similarity

Lesson 6-6: Understand Similarity

In this Grade 7 lesson from Reveal Math, Accelerated (Unit 6: Congruence and Similarity), students learn to define similar figures as shapes with proportional corresponding sides and the same shape but not necessarily the same size. Students practice identifying similarity by comparing ratios of side lengths and by describing sequences of transformations — including dilations, rotations, and translations — that map one figure onto another. The lesson connects scale factor to real-world contexts such as architectural drawings and video game programming.

Section 1

Defining Similarity (Angles and Proportions)

Property

In geometry, "similar" (denoted by $\sim$ ) has a very strict mathematical meaning. Two figures are similar if they have the exact same shape, but not necessarily the same size. For this to be true, two rules must be met simultaneously:

All corresponding angles must be exactly congruent (equal).
All corresponding side lengths must be proportional (they share the same scale factor, $k$ ).

Examples

The Blueprint: A floor plan and the actual house are similar. If a room's corner is $90^\circ$ on the paper, it must be exactly $90^\circ$ in the real house.
Finding Missing Sides: $\Delta ABC \sim \Delta XYZ$ . If side $AB = 4$ and $BC = 6$ , and the corresponding side $XY = 8$ , we know the scale factor is $2$ (because $8 / 4 = 2$ ). Therefore, side $YZ$ must be $6 \times 2 = 12$ .

Explanation

Think of similar shapes as perfect zooming in or out. The angles act as the "skeleton" that keeps the shape from distorting, which is why angles NEVER change during a zoom. The sides act as the "muscles" that stretch or shrink proportionally. If one side doubles in length, every other side must also double, otherwise the shape warps and is no longer similar!

Section 2

Testing Similarity with Proportions

Property

To officially prove two polygons are similar without using transformations, you must test the ratios of ALL their corresponding sides. You create a fraction for each pair of sides: $\frac{\text{Image Side}}{\text{Pre-image Side}}$ . If every single fraction simplifies to the exact same number (the scale factor $k$ ), the figures are similar.

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

Examples

Testing Triangles: Triangle 1 has sides $3, 4, 5$ . Triangle 2 has sides $6, 8, 10$ .
- Check ratios: $\frac{6}{3} = 2$ , $\frac{8}{4} = 2$ , $\frac{10}{5} = 2$ . All equal $2$ . They are similar!
Testing Rectangles: Rectangle A is $4$ by $6$ . Rectangle B is $6$ by $8$ .
- Check ratios: $\frac{6}{4} = 1.5$ , but $\frac{8}{6} \approx 1.33$ . The ratios are NOT equal. They are not similar.

Explanation

This is the ultimate test for similarity. A common mistake students make is only checking one pair of sides and assuming the whole shape is similar. You must check every pair! When setting up your fractions, be organized: always put the sides of the second shape on the top (numerator), and the sides of the first shape on the bottom (denominator). Make sure you are matching the shortest side to the shortest side, and the longest to the longest!

Section 3

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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Similarity (Angles and Proportions)

Property

All corresponding angles must be exactly congruent (equal).
All corresponding side lengths must be proportional (they share the same scale factor, $k$ ).

Examples

The Blueprint: A floor plan and the actual house are similar. If a room's corner is $90^\circ$ on the paper, it must be exactly $90^\circ$ in the real house.
Finding Missing Sides: $\Delta ABC \sim \Delta XYZ$ . If side $AB = 4$ and $BC = 6$ , and the corresponding side $XY = 8$ , we know the scale factor is $2$ (because $8 / 4 = 2$ ). Therefore, side $YZ$ must be $6 \times 2 = 12$ .

Explanation

Section 2

Testing Similarity with Proportions

Property

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

Examples

Testing Triangles: Triangle 1 has sides $3, 4, 5$ . Triangle 2 has sides $6, 8, 10$ .
- Check ratios: $\frac{6}{3} = 2$ , $\frac{8}{4} = 2$ , $\frac{10}{5} = 2$ . All equal $2$ . They are similar!
Testing Rectangles: Rectangle A is $4$ by $6$ . Rectangle B is $6$ by $8$ .
- Check ratios: $\frac{6}{4} = 1.5$ , but $\frac{8}{6} \approx 1.33$ . The ratios are NOT equal. They are not similar.

Explanation

Section 3

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

Overview

Lesson overview

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

Overview

Section 1

Defining Similarity (Angles and Proportions)

Property

All corresponding angles must be exactly congruent (equal).
All corresponding side lengths must be proportional (they share the same scale factor, $k$ ).

Examples

The Blueprint: A floor plan and the actual house are similar. If a room's corner is $90^\circ$ on the paper, it must be exactly $90^\circ$ in the real house.
Finding Missing Sides: $\Delta ABC \sim \Delta XYZ$ . If side $AB = 4$ and $BC = 6$ , and the corresponding side $XY = 8$ , we know the scale factor is $2$ (because $8 / 4 = 2$ ). Therefore, side $YZ$ must be $6 \times 2 = 12$ .

Explanation

Section 2

Testing Similarity with Proportions

Property

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k

Examples

Testing Triangles: Triangle 1 has sides $3, 4, 5$ . Triangle 2 has sides $6, 8, 10$ .
- Check ratios: $\frac{6}{3} = 2$ , $\frac{8}{4} = 2$ , $\frac{10}{5} = 2$ . All equal $2$ . They are similar!
Testing Rectangles: Rectangle A is $4$ by $6$ . Rectangle B is $6$ by $8$ .
- Check ratios: $\frac{6}{4} = 1.5$ , but $\frac{8}{6} \approx 1.33$ . The ratios are NOT equal. They are not similar.

Explanation

Section 3

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.