Similar Figures and Scale Factors

In geometry, two figures are similar if they have the exact same shape but not necessarily the same size. Think of it like zooming in or out on a photograph.

For two polygons to be similar, they must meet two conditions:

1. All corresponding angles are exactly equal.
2. All corresponding sides are proportional (they share a constant ratio).

What is a Scale Factor?

The scale factor (often represented by the letter $k$) is the ratio of the lengths of corresponding sides of two similar figures.

$k = \frac{\text{Side length of the new figure}}{\text{Corresponding side length of original figure}}$

If $k > 1$, the figure is enlarged. If $0 < k < 1$, the figure is reduced.

Scaling Perimeters and Areas

When you scale a figure by a factor of $k$, not everything scales by that exact same number. The dimension of the measurement determines the rule:

• 1D Measurements (Lengths and Perimeters): Scale by exactly $k$. The ratio of the perimeters of two similar figures is the same as the scale factor of their sides.
• 2D Measurements (Area): Scale by $k^2$. Because area involves multiplying two 1D lengths (like base and height), the scale factor is applied twice.

Example Problems

Example 1: Finding a missing side length If $\triangle ABC \sim \triangle DEF$ with a scale factor of $3:5$, and $AB = 9$, find $DE$.

Solution: The scale factor tells us the ratio of corresponding sides is $\frac{3}{5}$. We can set up a proportion: $\frac{AB}{DE} = \frac{3}{5}$ Substitute the known value of $AB$: $\frac{9}{DE} = \frac{3}{5}$ Cross-multiply to solve for $DE$: $3 \cdot DE = 9 \cdot 5$ $3 \cdot DE = 45$ $DE = 15$

Example 2: Finding a missing area If two similar rectangles have a scale factor of $2:3$, and the smaller rectangle has an area of $24$, find the area of the larger rectangle.

Solution: The ratio of their corresponding sides is $\frac{2}{3}$. Because area scales by the square of the scale factor, the ratio of their areas will be: $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ Let $A$ be the area of the larger rectangle. Set up the area proportion: $\frac{\text{Area of smaller}}{\text{Area of larger}} = \frac{4}{9}$ $\frac{24}{A} = \frac{4}{9}$ Cross-multiply to solve for $A$: $4 \cdot A = 24 \cdot 9$ $4 \cdot A = 216$ $A = 54$ The area of the larger rectangle is $54$.