Lesson 4: Similarity

New Concept

In mathematics, 'similar' figures have the same shape but can differ in size. We'll learn the two rules for similarity: corresponding angles are equal, and corresponding sides are proportional, linked by a constant 'scale factor'.

What’s next

This is just the beginning! Next, you'll tackle interactive examples and practice problems to master finding scale factors and unknown side lengths.

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.

Property

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

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).

Property

Two figures are called proportional if the ratios of corresponding distances are equal. For similar figures, the ratio of any two corresponding side lengths is equal to the scale factor.

Examples

• A triangle with sides 3, 5, and 7 is proportional to a triangle with sides 6, 10, and 14 because all side ratios equal 2 ($\frac{6}{3} = \frac{10}{5} = \frac{14}{7} = 2$).
• A 4x6 photo is enlarged to a 10x15 poster. The figures are proportional because the ratios of corresponding sides are equal: $\frac{10}{4} = 2.5$ and $\frac{15}{6} = 2.5$.
• A rectangle with sides 5 and 8 is not proportional to a rectangle with sides 10 and 18, because the ratios of corresponding sides are not equal ($\frac{10}{5} = 2$ but $\frac{18}{8} = 2.25$ ).

Explanation

Proportional means that all corresponding parts of two figures are in the same ratio. If one side of a figure is twice as long as its corresponding side on a similar figure, then all other corresponding sides will also be twice as long.

Property

Two figures are similar if, and only if:

• Their corresponding angles are equal, and
• Their corresponding sides are proportional.

Both conditions must be true for figures to be similar.

Examples

• A square and a non-square rhombus both have proportional sides, but their angles are not equal. Therefore, they are not similar.
• A 3x5 rectangle and a 6x10 rectangle are similar. All their angles are $90^\circ$ (equal), and their corresponding sides are proportional ($\frac{6}{3} = \frac{10}{5} = 2$).
• Two trapezoids are similar. One has bases of length 4 and 10 and a height of 3. If the similar trapezoid has a height of 9, its scale factor is $\frac{9}{3}=3$. Its bases will be $4 \times 3=12$ and $10 \times 3=30$.

Explanation

For any two shapes to be similar, they must pass two tests. First, all their matching angles must be equal. Second, the ratios of all their matching sides must be the same. Failing either test means they are not similar.

Property

Two triangles are similar if either one of the following conditions is true:

1. Their corresponding angles are equal.
2. Their corresponding sides are proportional.

Examples

• One triangle has angles $30^\circ$ and $70^\circ$. Another has angles $70^\circ$ and $80^\circ$. The third angle in the first is $80^\circ$ and in the second is $30^\circ$. Since all three corresponding angles are equal, the triangles are similar.
• A triangle has sides of length 5, 12, and 13. A second triangle has sides 10, 24, and 26. The ratios of corresponding sides are all equal to 2 ($\frac{10}{5} = \frac{24}{12} = \frac{26}{13} = 2$), so they are similar.
• A large triangle is formed by a 20-foot flagpole and its 15-foot shadow. A person who is 6 feet tall stands nearby. Their shadow forms a smaller, similar triangle. The length of the person's shadow, $s$, can be found by the proportion $\frac{6}{s} = \frac{20}{15}$, so $s=4.5$ feet.

Explanation

Triangles have a special shortcut for similarity. You only need to prove one of the two conditions. If their angles match, their sides must be proportional. If their sides are proportional, their angles must match. This makes them easier to work with.