# Problem Solving in Geometry with Proportions

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Problem Solving in Geometry with Proportions

IF a b a c , then = = c d b d IF a c a + b c + d , then = = b d b d Slide #2

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 Slide #3

Ex. 1: Using Properties of Proportions
IF p 3 = Simplify r 5  The statement is true. Slide #4

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 Slide #5

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? Slide #6

Cross Product Property Divide each side by 20.
Solution AB = AC BD CE 16 = 30 – 10 x 16 = 20 x 20x = 160 x = 8 Given Substitute Simplify Cross Product Property Divide each side by 20. So, the length of BD is 8. Slide #7

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 Slide #8

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 Slide #9

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. Slide #10

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 Slide #11

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. Slide #12

Ex. 4: Solving a proportion
MODEL BUILDING. A scale model of the Titanic is inches long and inches wide. The Titanic itself was 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 = in Length of Titanic = feet Length of model ship = in. Slide #13

Reasoning: = = = Write the proportion. Substitute.
Multiply each side by Use a calculator. Width of Titanic Length of Titanic = Width of model Length of model x feet 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. Slide #14

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. Slide #15