Geometry 6-1

Big Idea: Use Ratios & Proportions

Ratio A comparison of two A comparison of two numbersnumbersEx.1) ½, 1:2

Ex.2) 3 , 3:4 4

Example 7 weeks: 14 days(convert to the same units in order

to compare, then simplify) Method 1: 49 days = 7 (reduce) 14 days 2 Method 2: 7 weeks = 7 2 weeks 2

Proportions

An equation that states that 2 ratios An equation that states that 2 ratios are equal.are equal.

1 = 2 (Read as:

2 4 “1 is to 2 as 2 is to 4”)

1 = 2

2 4

extremes & means(extremes are the first and last terms)

(means are the middle terms)

Property of Cross Products of a Proportion:

the product of the means always equals the product of the extremes

Ex.1) 1 = 2

2 4 (1·4 = 2·2)

Ex.2) 3 = 6_ (3·14 = 7·6)

7 14

Example: Solve the proportion

Example: solve the proportion

Example: Find the dimensions (l, w) of a wall whose perimeter is 484 m and whose ratio of length to width is 9:2.

An extended ratio compares more than 2 numbers.

Example: The measures of the angles of a triangle are in the extended ratio of 3:4:8. Find the angle measures.

The geometric mean is the positive number, if placed in the position of the means makes the proportion a true statement.

The geometric mean of 2 positive numbers (a & b) is “x” where

a = x x b Then by the rule of cross-products,

x(x) = ab and x2 = ab

√x2 = √ab x = √ab

Example: Find the geometric mean of 32 & 8.

Example: Find the geometric mean of 16 & 18.