Problem Solving in Geometry with Proportions

Ratio A ratio is a comparison of two quantities by division The ratio of a and b can be represented three ways: a/b a:b a to b An extended ration compares three values (i.e. a:b:c)

Ratio Example

Extended Ratio Example

Proportion A Proportion is an equation that states two proportions are equal First and last numbers are the Extremes Middle two numbers are the Means

Cross Products Property

Additional Properties of Proportions IF a b a c , then = = c d b d IF a c a + b c + d , then = = b d b d

Ex. 1: Using Properties of Proportions IF p 3 p r , then = = r 5 6 10 p r Given = 6 10 p 6 a c a b = = = , then b d c d r 10

Ex. 1: Using Properties of Proportions IF p 3 = Simplify r 5  The statement is true.

Ex. 1: Using Properties of Proportions a c Given = 3 4 a + 3 c + 4 a c a + b c + d = = = , then 3 4 b d b d Because these conclusions are not equivalent, the statement is false. a + 3 c + 4 ≠ 3 4

Ex. 2: Using Properties of Proportions In the diagram AB AC = BD CE Find the length of BD. Do you get the fact that AB ≈ AC?

Cross Product Property Divide each side by 20. Solution AB = AC BD CE 16 = 30 – 10 x 10 16 = 20 x 10 20x = 160 x = 8 Given Substitute Simplify Cross Product Property Divide each side by 20. So, the length of BD is 8.

a x x b Geometric Mean = The geometric mean of two positive numbers a and b is the positive number x such that a x If you solve this proportion for x, you find that x = √a ∙ b which is a positive number. = x b

Geometric Mean Example For example, the geometric mean of 8 and 18 is 12, because 8 12 = 12 18 and also because x = √8 ∙ 18 = x = √144 = 12

Ex. 3: Using a geometric mean PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.

Write proportion 210 x = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 Solution: The geometric mean of 210 and 420 is 210√2, or about 297mm. 210 x Write proportion = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 X = √210 ∙ 210 ∙ 2 X = 210√2 Cross product property Simplify Factor Simplify

Using proportions in real life In general when solving word problems that involve proportions, there is more than one correct way to set up the proportion.

Ex. 4: Solving a proportion MODEL BUILDING. A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it? Width of Titanic Length of Titanic = Width of model Length of model LABELS: Width of Titanic = x Width of model ship = 11.25 in Length of Titanic = 882.75 feet Length of model ship = 107.5 in.

Reasoning: = = = Write the proportion. Substitute. Multiply each side by 11.25. Use a calculator. Width of Titanic Length of Titanic = Width of model Length of model x feet 882.75 feet = 11.25 in. 107.5 in. 11.25(882.75) x = 107.5 in. x ≈ 92.4 feet So, the Titanic was about 92.4 feet wide.

Note: Notice that the proportion in Example 4 contains measurements that are not in the same units. When writing a proportion in unlike units, the numerators should have the same units and the denominators should have the same units. The inches (units) cross out when you cross multiply.