Ratios and Proportions Keystone Geometry

Ratio A ratio is a comparison of two numbers such as a : b. Ratio: When writing a ratio, always express it in simplest form. ** Ratios must be compared using the same units. A ratio can be expressed: 1. As a fraction 2. As a ratio 3 : 7 3. Using the word “to” 3 to 7

Example: What is the ratio of side AB to side CB in the triangle? 3.6 6 8 4.8 10 Now try to reduce the fraction. Example: What is the ratio of side DB to side CD in the triangle? Type notes here

Example ………. A baseball player goes to bat 348 times and gets 107 hits. What is the players batting average? Solution: Set up a ratio that compares the number of hits to the number of times he goes to bat. Ratio: Convert this fraction to a decimal rounded to three decimal places. Decimal: Type notes here The baseball player’s batting average is 0.307 which means he is getting approximately one hit every three times at bat.

Proportion Definition: A proportion is an equation stating that two ratios are equal. For example,

Terms of a Proportion First Term Third Term Second Term Fourth Term Type notes here Third Term Second Term

Means and Extremes The first and last terms of a proportion are called extremes. The middle terms are called the means. ** The product of the means is always equal to the product of the extremes.

Properties of Proportions is equal to: Cross-multiplication Switching the means Add one to both sides Reciprocals

Example: If , then… 2x 5y = _____

** Special Note: The easiest way to decide if two proportions are equal is to apply the mean-extremes property (cross multiplication). However, all of the other properties work as well, provided your initial proportion is true.

Proportions- examples…. Solve the proportion using cross multiplication. 4 • x = 12 • 3 4x = 36 x = 9 4x = 36 4 4

Some to try… 1. 2. 3. 4.

Example 2: Use a proportion to solve for the missing piece of a triangle. 84 yards 2 ft x 356 yards Find the value of x. First! Multiply by 3 to change yards into feet. The units must be the same. Type notes here

Examples: Find the measure of each angle. Two complementary angles have measures in the ratio 2 : 3. Two supplementary angles have measures in the ratio 3 : 7. The measures of the angles of a triangle are in a ratio of 2 : 2 : 5. The perimeter of a triangle is 48cm and the lengths of the sides are in a ratio of 3 : 4 : 5. Find the length of each side. 36 and 54 54 and 126 40, 40, and 100 12cm, 16cm, and 20cm