Lesson 5: Similar Figures

Property

Two figures are similar if, and only if:

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

Examples

• A rectangle with sides 4 cm and 6 cm is similar to a rectangle with sides 8 cm and 12 cm. The scale factor is 2.

• A photograph measuring 4 inches by 6 inches is enlarged to 8 inches by 12 inches. The enlarged photo is similar to the original.

Property

The symbol $\sim$ means "is similar to" and is used to write similarity statements between figures. When writing similarity statements like $\triangle ABC \sim \triangle DEF$, the order of vertices must match the correspondence between the figures.

Examples

Property

The principle of similar figures is used to solve problems involving indirect measurement. Common applications include using shadows to find the height of tall objects or using a map's scale to find actual distances. The method involves setting up a proportion where the ratios of corresponding sides of the similar figures (e.g., triangles formed by objects and their shadows) are equal.

Examples

• A 5-foot-tall person casts a 4-foot shadow. A nearby flagpole casts a 20-foot shadow. How tall is the flagpole? Let $h$ be the height. $\dfrac{5 \text{ ft}}{4 \text{ ft}} = \dfrac{h \text{ ft}}{20 \text{ ft}} \implies 4h = 100 \implies h=25$. The pole is 25 feet tall.
• On a map, the distance between two cities is 3 inches. The map scale says 0.5 inches represents 40 miles. What is the actual distance? Let $d$ be the distance. $\dfrac{0.5 \text{ in}}{40 \text{ miles}} = \dfrac{3 \text{ in}}{d \text{ miles}} \implies 0.5d = 120 \implies d=240$ miles.
• A 10-foot street lamp casts a 15-foot shadow. A nearby fire hydrant casts a 3-foot shadow. What is the hydrant's height? Let $h$ be the height. $\dfrac{10 \text{ ft}}{15 \text{ ft}} = \dfrac{h \text{ ft}}{3 \text{ ft}} \implies 15h = 30 \implies h=2$. The hydrant is 2 feet tall.

Explanation

Can't measure a tall tree? Use its shadow! An object of known height and its shadow form a triangle.