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

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: ${\tfrac {12}{16}}$ = ${\tfrac {6}{8}}$ = ${\tfrac {3}{4}}$ All proportions have 4 parts known as the extremes and means. In ${\tfrac {6}{8}}$ = ${\tfrac {3}{4}}$ , the extremes are $6$ and $4$ , while the means are $8$ and $3$ . The Cross-Product Property states that the products of the extremes and means are equal. So:

• $4$ $6$ = $24$ • $8$ $3$ = $24$ You can use the cross-product property to check if two ratios are a proportion.

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

 Yes No

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

 Yes No

## Solving with $x$ How do we solve porpotions with $x$ or any variable?

### Problem #1

Solve the proportion

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

$3x=42$ And simply work out the problem from there:

$3x=42$ ${\tfrac {3x}{3}}$ = ${\tfrac {42}{3}}$ $x=14$ So, alas, the $x$ in ${\tfrac {x}{6}}$ = ${\tfrac {7}{3}}$ is $14$ ... So: ${\tfrac {14}{6}}$ = ${\tfrac {7}{3}}$ 