Lesson 9-3: Similarity and Transformations

Property

Two polygons are similar if and only if there is a similarity transformation (a sequence of rigid motions and dilations) that maps one polygon exactly onto the other.

If polygon $A$ is mapped to polygon $B$ by a similarity transformation, then polygon $A \sim$ polygon $B$.

Property

Transformations are divided into two distinct categories.

Reflections, rotations, and translations are called congruence transformations (or isometries) because the original figure and its image remain perfectly congruent.

Dilations (resizing) are similarity transformations because the original figure and its image are similar, not congruent.

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).