Geometry

Ratios and Proportions

Big Picture

The use of ratios and proportions come sin handy when finding the relationship between two geometric shapes. Ratios and proportions are often an important part of certain properties between two shapes, such as two similar polygons.

Key Terms

Ratio: A way to compare two numbers.

Extended Ratio: A way to compare three or more numbers.

Proportion: An equation that has two ratios equal to each other.

Scale: A ratio that relates the dimensions in a drawing to the actual dimensions of an object.

Using Ratios and Proportions

A ratio can be written in three ways (y ≠ 0):

  1. \(\frac{x}{y}\)
  2. x:y
  3. x to y
  • Ratios are usually written in simplest form.
  • It is easier to simplify ratios in fraction form.
  • Ratios with the same simplified form are equivalent.
  • Example: 7:14 and 1:2 are equivalent ratios.
  • Simplified ratios should not have units.
  • The numerator and denominator must have the same units before simplifying - multiply by a conversion factor if needed.
  • Example of conversion factor: \(\frac{12 in}{1 ft}\)

An extended ratio of three numbers is written as x:y:z.

 The order of the numbers in a ratio matter!

A proportion relates two equivalent ratios: \(\frac{\color{red}{a}}{\color{blue}{b}} \color{black}{=} \frac{\color{blue}{c}}{\color{red}{d}}\color{black}{,} \color{blue}{b} \color{black}{\text{ and }} \color{red}{d} \color{black}{\neq} 0\)

There are four parts to the proportion (\(\color{red}{a}, \color{blue}{b}, \color{blue}{c}, \color{red}{d}\))

  • \(\color{blue}{b}\), \(\color{blue}{c}\) are called the means
  • \(\color{red}{a}\), \(\color{red}{d}\) are called the extremes

Scale

Ratios and proportions are used in maps and scale drawings.

  • In a scale drawing, the drawing has the same shape but different size as the object it represents.
  • Similarly, a map accurately represents real-life distances at a more convenient size.

Maps and scale drawings typically have a scale that relates the dimensions in a drawing to the actual dimensions.

  • Scale = Drawing dimension: actual dimension
  • A  scale can  have units  (e.g. 1  cm to 1 km) or  be simplified to not have units.

(e.g.\(\frac{1 cm}{1 km} = \frac{1 cm}{10^5 cm} = \frac{1}{10,000}\))

Using the scale, a proportion can be set up to find the actual dimensions.

  • Actual dimension = Scale • drawing dimension

Never assume diagrams are drawn to scale - rely on measurements and markings in the diagram.

Cross-Multiplication

Cross-Multiplication Theorem: If \(\frac{a}{b} = \frac{c}{d}\), where a, b, c, and d are real numbers and b and d are not equal to 0, then

the product of the extremes equals the product of the means: ad = bc.

Corollaries

The following manipulations of a proportion will still make it true:

  • Swapping the means: If a, b, c, and d are nonzero and \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a}{c} = \frac{b}{d}\).
  • Swapping the extremes: If a, b, c, and d are nonzero \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{d}{b} = \frac{c}{a}\).
  • Taking the reciprocal (flipping it upside down): If a, b, c, and d are nonzero and \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{b}{a} = \frac{d}{c}\).
  • If a, b, c, and d are nonzero and \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a+b}{b} = \frac{c+d}{d}\).
  • If a, b, c, and d are nonzero and \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a-b}{b} = \frac{c-d}{d}\).