How to Find Similarity and Ratios? (+FREE Worksheet!)

When two figures are similar, their angles are equal and their corresponding side lengths form equal ratios called a scale factor. Understanding similarity and ratios lets you find missing side lengths, calculate scale distances on maps, and solve many real-world geometry problems in Algebra 1 and beyond.

What Is Similarity?

Two figures are similar (~) if they have the same shape but not necessarily the same size. Similar figures have:

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• All corresponding angles equal.
• All corresponding side lengths in the same ratio (the scale factor).

For example, if triangle ABC ~ triangle DEF, then \(\color{blue}{\frac{\text{ AB }}{\text{ DE }} = \frac{\text{ BC }}{\text{ EF }} = \frac{\text{ AC }}{\text{ DF }}}\).

Key Concepts: Scale Factor and Ratios

Scale Factor

The scale factor is the ratio of any pair of corresponding side lengths. If triangle 1 has sides 4, 6, and 8 and triangle 2 has sides 12, 18, and 24, the scale factor is \(\color{blue}{\frac{4}{12} = \frac{1}{3}}\) (or 3 if going the other direction).

Setting Up the Proportion

To find a missing side, set corresponding sides equal as a proportion and solve with cross-multiplication.

• If \(\color{blue}{\frac{5}{10} = \frac{x}{14}}\), cross-multiply: \(\color{blue}{10x = 70}\), so \(\color{blue}{x = 7}\).
• If \(\color{blue}{\frac{8}{12} = \frac{6}{x}}\), cross-multiply: \(\color{blue}{8x = 72}\), so \(\color{blue}{x = 9}\).

Simplifying the Ratio

Always simplify side ratios to confirm similarity. A ratio of 6:9 simplifies to \(\color{blue}{2:3}\); a ratio of 8:12 also simplifies to \(\color{blue}{2:3}\), confirming the two pairs of sides are proportional.

Step-by-Step Summary

1. Identify the corresponding sides of the two similar figures.
2. Write the ratio of a known pair of corresponding sides to find the scale factor.
3. Set up a proportion using the scale factor and the unknown side.
4. Cross-multiply and solve for the unknown.
5. Check by verifying all ratios of corresponding sides are equal.

Watch: Equivalent Ratios in Similar Shapes

Khan Academy shows how equivalent ratios arise from similar shapes in geometry:

Similarity and Ratios – Worked Examples

Example 1: Two similar rectangles have widths 4 cm and 12 cm. The length of the smaller rectangle is 6 cm. Find the length of the larger rectangle.

Scale factor: \(\color{blue}{\frac{12}{4} = 3}\). Larger length = \(\color{blue}{6 \times 3 = 18 \text{ cm }}\).

Example 2: Two similar triangles have sides in ratio 2:3. If the shorter triangle has a side of 8, find the corresponding side of the larger triangle.

Set up: \(\color{blue}{\frac{2}{3} = \frac{8}{x}}\). Cross-multiply: \(\color{blue}{2x = 24}\), so \(\color{blue}{x = 12}\).

Example 3: Solve for x: \(\color{blue}{\frac{5}{10} = \frac{x}{14}}\).

Cross-multiply: \(\color{blue}{10x = 70}\), so \(\color{blue}{x = 7}\).

Example 4: Solve for x: \(\color{blue}{\frac{8}{12} = \frac{6}{x}}\).

Cross-multiply: \(\color{blue}{8x = 72}\), so \(\color{blue}{x = 9}\).

More Practice: Using Similarity to Estimate Ratios

Khan Academy demonstrates how to use similarity to estimate the ratio between side lengths:

Exercises for Similarity and Ratios

Find the missing side length in each pair of similar figures.

1. \(\color{blue}{\frac{3}{6} = \frac{x}{10}}\)
2. \(\color{blue}{\frac{4}{6} = \frac{8}{x}}\)
3. \(\color{blue}{\frac{5}{15} = \frac{x}{9}}\)
4. \(\color{blue}{\frac{10}{15} = \frac{6}{x}}\)
5. Two similar triangles have sides in ratio 3:5. The smaller has a side of 12. Find the corresponding larger side.
6. A scale model is built at a 1:20 ratio. If a wall on the model is 5 cm, how long is the real wall in cm?

Answers

1. \(\color{blue}{x = 5}\)
2. \(\color{blue}{x = 12}\)
3. \(\color{blue}{x = 3}\)
4. \(\color{blue}{x = 9}\)
5. \(\color{blue}{20}\) (set up \(\color{blue}{\frac{3}{5} = \frac{12}{x}}\); \(\color{blue}{x = 20}\))
6. \(\color{blue}{100 \text{ cm }}\) (set up \(\color{blue}{\frac{1}{20} = \frac{5}{x}}\); \(\color{blue}{x = 100}\))

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Frequently Asked Questions

What is the difference between congruent and similar figures?

Congruent figures have exactly the same shape and size (scale \(\color{blue}{\text{ factor } = 1}\)). Similar figures have the same shape but may differ in size (any scale factor).

Do all three pairs of sides need to be proportional for triangles to be similar?

If you can confirm two pairs of corresponding angles are equal (AA similarity), the triangles are similar and all three side ratios will automatically be equal.

Can similar figures be rotated or reflected?

Yes. Similarity allows rotations, reflections, and translations in addition to scaling. Match corresponding vertices carefully before setting up your proportion.

Related Topics

• Proportional Ratios
• Percent Problems
• Order of Operations
• Discount, Tax, and Tip
• Simple Interest