# Geometry/Chapter 5/Lesson 2

## Introduction

We will be reviewing proportions, the properties of proportions, and how to solve them.

## Proportions

A proportion is a statement that states that two ratios are equal.

EXAMPLE: ${\displaystyle {\tfrac {6}{8}}}$ and ${\displaystyle {\tfrac {3}{4}}}$ are equal, and thus are a proportion: ${\displaystyle {\tfrac {6}{8}}}$ = ${\displaystyle {\tfrac {3}{4}}}$.

An extended proportion is similar to an extended ration from the last lesson: A statement that states that three or more rations are equal.

EXAMPLE: ${\displaystyle {\tfrac {12}{16}}}$ = ${\displaystyle {\tfrac {6}{8}}}$ = ${\displaystyle {\tfrac {3}{4}}}$

All proportions have 4 parts known as the extremes and means. In ${\displaystyle {\tfrac {6}{8}}}$ = ${\displaystyle {\tfrac {3}{4}}}$, the extremes are ${\displaystyle 6}$ and ${\displaystyle 4}$, while the means are ${\displaystyle 8}$ and ${\displaystyle 3}$. The Cross-Product Property states that the products of the extremes and means are equal. So:

• ${\displaystyle 4}$${\displaystyle 6}$ = ${\displaystyle 24}$
• ${\displaystyle 8}$${\displaystyle 3}$ = ${\displaystyle 24}$

You can use the cross-product property to check if two ratios are a proportion.

1 Are ${\displaystyle {\tfrac {3}{9}}}$ = ${\displaystyle {\tfrac {5}{7}}}$ a proportion?

 Yes No

2 Are ${\displaystyle {\tfrac {9}{3}}}$ = ${\displaystyle {\tfrac {12}{4}}}$ a proportion?

 Yes No

## Solving with ${\displaystyle x}$

How do we solve porpotions with ${\displaystyle x}$ or any variable?

### Problem #1

Solve the proportion

${\displaystyle {\tfrac {x}{6}}}$ = ${\displaystyle {\tfrac {7}{3}}}$

Solving for ${\displaystyle x}$, you would multiply the extremes (${\displaystyle 3}$ and ${\displaystyle x}$) and the means (${\displaystyle 6}$ and ${\displaystyle 7}$):

${\displaystyle 3x=42}$

And simply work out the problem from there:

${\displaystyle 3x=42}$

${\displaystyle {\tfrac {3x}{3}}}$ = ${\displaystyle {\tfrac {42}{3}}}$

${\displaystyle x=14}$

So, alas, the ${\displaystyle x}$ in ${\displaystyle {\tfrac {x}{6}}}$ = ${\displaystyle {\tfrac {7}{3}}}$ is ${\displaystyle 14}$... So: ${\displaystyle {\tfrac {14}{6}}}$ = ${\displaystyle {\tfrac {7}{3}}}$