8.2 Problem Solving in Geometry with Proportions

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8.2 Problem Solving in Geometry with Proportions. Geometry Mrs. Spitz Spring 2005. Use properties of proportions Use proportions to solve real-life problems such as using the scale of a map. Pp. 2-32 all Reminder: Quiz after 8.3. Ch. 8 Definitions due Ch. 8 Postulates/Theorems due.

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8.2 Problem Solving in Geometry with Proportions

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8.2 Problem Solving in Geometry with Proportions

Geometry

Mrs. Spitz

Spring 2005

Use properties of proportions

Use proportions to solve real-life problems such as using the scale of a map.

Pp. 2-32 all

Reminder: Quiz after 8.3.

Ch. 8 Definitions due

Ch. 8 Postulates/Theorems due

Objectives/Assignment

Slide #2

Using the properties of proportions
• In Lesson 8.1, you studied the reciprocal property and the cross product property. Two more properties of proportions, which are especially useful in geometry, are given on the next slides.
• You can use the cross product property and the reciprocal property to help prove these properties.

Slide #3

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

Slide #4

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 #5

Ex. 1: Using Properties of Proportions

IF

p

3

=

Simplify

r

5

 The statement is true.

Slide #6

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 #7

In the diagramEx. 2: Using Properties of Proportions

AB

AC

=

BD

CE

Find the length of

BD.

Do you get the fact that AB ≈ AC?

Slide #8

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.

Slide #9

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 #10

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 #11

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.Ex. 3: Using a geometric mean

Slide #12

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 #13

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 #14

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.

Slide #15

Write the proportion.

Substitute.

Multiply each side by 11.25.

Use a calculator.

Reasoning:

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.

Slide #16

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 #17