Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 1: Transformations

Lesson 5: Similar Figures

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, students learn to identify similar figures by naming corresponding angles and corresponding sides, and to determine whether figures are similar by testing if their side lengths are proportional. Students also practice setting up and solving proportions to find unknown measures of similar figures, using real-world contexts like resizing photographs and creating scaled designs.

Section 1

Introduction to Similar Figures

Property

Two figures are called similar if they have the same shape but different sizes. In similar figures:

The corresponding angles are equal.
We can multiply each side of one figure by the same factor (the scale factor) to get the corresponding side of the other figure.

Examples

Two triangles both have angles $45^\circ$ , $45^\circ$ , and $90^\circ$ . Because their corresponding angles are equal, they are similar, regardless of their side lengths.
A rectangle with sides 4 cm and 6 cm is not similar to a square with sides 4 cm. Although they share a side length, their overall shapes and side ratios are different.
A circle with a radius of 5 units and a circle with a radius of 15 units are similar. All circles have the same shape.

Explanation

Think of similar figures as a photo and its enlargement. The shape is exactly the same, but the size is different. Every part of the figure is scaled up or down by the same amount, and all the angles remain identical.

Section 2

Scale Factors

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

\text{scale factor} = \frac{\text{new length}}{\text{corresponding original length}}

\text{new length} = \text{scale factor} \times \text{corresponding original length}

Examples

A map has a scale factor of $\frac{1}{10000}$ . A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life.
A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$ .
To reduce a 12-foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$ , its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.

Explanation

The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Introduction to Similar Figures

Property

Two figures are called similar if they have the same shape but different sizes. In similar figures:

The corresponding angles are equal.
We can multiply each side of one figure by the same factor (the scale factor) to get the corresponding side of the other figure.

Examples

Two triangles both have angles $45^\circ$ , $45^\circ$ , and $90^\circ$ . Because their corresponding angles are equal, they are similar, regardless of their side lengths.
A rectangle with sides 4 cm and 6 cm is not similar to a square with sides 4 cm. Although they share a side length, their overall shapes and side ratios are different.
A circle with a radius of 5 units and a circle with a radius of 15 units are similar. All circles have the same shape.

Explanation

Section 2

Scale Factors

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

\text{scale factor} = \frac{\text{new length}}{\text{corresponding original length}}

\text{new length} = \text{scale factor} \times \text{corresponding original length}

Examples

A map has a scale factor of $\frac{1}{10000}$ . A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life.
A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$ .
To reduce a 12-foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$ , its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.

Explanation

Overview

Lesson overview

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

Overview

Section 1

Introduction to Similar Figures

Property

Two figures are called similar if they have the same shape but different sizes. In similar figures:

The corresponding angles are equal.
We can multiply each side of one figure by the same factor (the scale factor) to get the corresponding side of the other figure.

Examples

Two triangles both have angles $45^\circ$ , $45^\circ$ , and $90^\circ$ . Because their corresponding angles are equal, they are similar, regardless of their side lengths.
A rectangle with sides 4 cm and 6 cm is not similar to a square with sides 4 cm. Although they share a side length, their overall shapes and side ratios are different.
A circle with a radius of 5 units and a circle with a radius of 15 units are similar. All circles have the same shape.

Explanation

Section 2

Scale Factors

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

\text{scale factor} = \frac{\text{new length}}{\text{corresponding original length}}

\text{new length} = \text{scale factor} \times \text{corresponding original length}

Examples

A map has a scale factor of $\frac{1}{10000}$ . A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life.
A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$ .
To reduce a 12-foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$ , its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.