Section 9.2: Proportionality and Similarity

Welcome to the notes for Section 9.2, where we'll delve into the fascinating world of proportionality and similarity in geometry. Get ready to explore how changing the scale of a figure affects its properties and discover some beautiful mathematical relationships.

Understanding Proportionality

A variable quantity $A$ is directly proportional to another quantity $B$ if it's always a fixed (nonzero) constant multiple of the other. This can be expressed mathematically as:

$$A = cB$$

Here, $c$ is the constant of proportionality. Think of it as the unchanging factor that links the two quantities. For example, if you drive at a constant speed of 60 miles per hour, the distance you travel is proportional to the time you spend driving. The equation is Distance = 60 × Time, where 60 is the constant of proportionality.

Properties of Proportionality

  • Two variable quantities are proportional when one is a (nonzero) constant multiple of the other.
  • If $A$ is proportional to $B$ with constant $c$, then $B$ is proportional to $A$ with constant $1/c$.
  • If one of two proportional quantities is multiplied by a certain factor, the other one is also multiplied by that factor. For instance, if $y$ is proportional to $x$, then multiplying $x$ by $k$ results in multiplying $y$ by $k$.
  • If one quantity doubles, the other will double as well.

The Golden Ratio

The golden ratio, often denoted by the Greek letter $\phi$ (phi), is approximately equal to 1.618. It's found throughout art, architecture, and nature. Mathematically, it's defined as the ratio where:

$$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.62$$

A golden rectangle is a rectangle in which the ratio of the length to the width is the golden ratio $\phi$. Therefore, among golden rectangles, the length is proportional to the width, and $\phi$ is the constant of proportionality.

Similar Triangles

Two triangles are similar if all corresponding angles have the same measure. A key property of similar triangles is that the ratios of corresponding sides are equal. If triangles ABC and DEF are similar, then:

$$\frac{A}{a} = \frac{B}{b} = \frac{C}{c}$$

Where A, B, and C are the sides of triangle ABC, and a, b, and c are the sides of triangle DEF.

Examples and Applications

Here are a couple examples to think about:

  • If the radius of a circle is doubled, what happens to the area? If the radius is $r$ and the area is $A$, then $A = \pi r^2$. If the radius is doubled to $2r$, the new area is $\pi (2r)^2 = 4\pi r^2 = 4A$. The area is multiplied by 4.
  • Suppose we have several boxes of different sizes, each of which is in the shape of a cube. Let $x$ denote the length of one side of the cube, and let $V$ denote the volume. Because the box is cubical in shape, the length, width, and height are all the same, $x$: $$V = Length \times Width \times Height = x \times x \times x = x^3$$ Thus, the volume is proportional to the cube of the length, and the constant of proportionality is 1.

Wrapping Up

We've covered the basics of proportionality and similarity, including the golden ratio and similar triangles. Remember to practice applying these concepts to various problems. Keep exploring, and you'll find these principles popping up in surprising places!