7.2 Measurement – Section 2 1 Objective A: To Convert between Celsius and Fahrenheit using the formulas There are two temperature scales: Fahrenheit, which

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7.2 Measurement – Section 2

Objective A: To Convert between Celsius and Fahrenheit using the formulas

9 5

F C 32 and C F 325 9

There are two temperature scales: Fahrenheit, which is used most often in the United States, and Celsius, which is used internationally and in science.

Example 1. Convert 30C to Fahrenheit

Solution: Replace C with 30 and evaluate.

9 30F 32

5 1

F 54 32 86 Answer: 86F

To convert form Celsius to Fahrenheit, use

9

F C 32.5

Your Turn Problem #1 Convert 25C to Fahrenheit.

Answer: 77F

2


When converting from Celsius to Fahrenheit, if the 5 does not divide evenly into the given value in Celsius:

To simplify, multiply the value of C by 9. Get that answer and divide by 5.

Example 2. Convert 43C to Fahrenheit

Solution: Replace C with 30 and evaluate.

9 43F 32

5 1

F 77.4 32 109.4

Answer: 109.4F

Your Turn Problem #2 Convert 12C to Fahrenheit.

Answer: 53.6F

387

F 325

3


To convert form Fahrenheit to Celsius, use 5

C F 32 .9

Example 3. Convert 59F to Celsius. If necessary round to the nearest tenth.

Solution: Replace F with 59 and evaluate.

5

C 59 329

5

C 279

Answer: 15C

Your Turn Problem #3 Convert 104F to Celsius.

Answer: 40C

We usually round to the nearest tenth if not exact.

4


Example 4. Convert 85F to Celsius. If necessary round to the nearest tenth.

Solution: Replace F with 85 and evaluate.

5

C 85 329

5

C 539

Your Turn Problem #4 Convert 100F to Celsius.

Answer: 37.8C

265

C 29.449

Answer: 29.4 C

5


An angle is a set of points consisting of two rays, or half-lines, with a common endpoint. The endpoint is called the vertex.

Objective B: An Introduction to Angles

Ray BC

Ray BA

CB

A

Vertex

The rays are called the sides. The angle to the left can be named angle ABC, angle CBA, or angle B.

Notice that the name of the vertex is either in the middle or, if no confusion results, listed by itself.

Next Slide

The symbol of angle is .We could also say: ABC, or CBA, or B.

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A device called a protractor is used to measure angles. Protractors have two scales.

To measure an angle like the angle shown below, we place the protractors (dot) at the vertex and line up one side of the angle’s sides at 0 degrees. Then we check where the angle’s other side crosses the scale.

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What is the measure of angle ABC?

Answer: MABC is 60 degrees.

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Types of Angles

Right angle: An angle that measures 90.Straight angle: An angle that measures 180.Acute angle: An angle that measures more than 0 but less than 90.Obtuse angle: An angle that measures more than 90 but less than 180

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When the sum of the measures of two angles is 90, the angles are said to be complementary.

1 2 90

65 25 90

m m65

25

1

2

Complementary Angles

Two angles are complementary if the sum of their measures is 90. Each angle is called a complement of the other.

Find the measure of the complement of a 42 angle.Example 5.

Solution: To find the complement of an angle, subtract it from 90.

Your Turn Problem #5

Find the measure of the complement of a 30 angle. Answer: 60

90 – 42 = 48

The measure of a complement is 48.

10


Two angles are supplementary if the sum of their measures is 180. Each angle is called a supplement of the other.

1 2 180

30 150 180

m m

150301

2

Supplementary angles

Example 6.Find the measure of a supplement of an angle of 122.

Solution:

180 – 122 = 58

The measure of a supplement is 58.122180 122

To find the supplement of an angle, subtract it from 180


Find the measure of the supplement of a 37 angle. Answer: 143

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A triangle is a polygon made up of three segments, or sides. We can classify triangles according to sides and according to angles.

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Equilateral triangle: All sides are the same length.Isosceles triangle: Two or more sides are the same length.Scalene triangle: All sides are of different lengths.Right triangle: One angle is a right angle.Obtuse triangle: One angle is an obtuse angle.Acute triangle: All three angles are acute.

Objective C: An Introduction to Triangles

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Given two of the angle measures of a triangle, find the third.

Sum of the Angle Measures of a Triangle

In any triangle ABC, the sum of the measures of the angles is 180:

180 .m A m B m C

Solution:

Example 7. Find the missing angle measure.

72 47

x

A B

C

180m A m B m C

72 4 x 17 80

119 x 180

x 180 119

x 61 The measure of angle C is 61.


Find the missing angle measure.110

25Answer:45

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Finding lengths of sides of similar triangles using proportions.

Look at the pair of triangles below. Note that they appear to have the same shape, but their sizes are different. These are examples of similar triangles.

Corresponding sides of similar triangles have the same ratio.

f d

e

a

d

cb

e f

c a

b

Similar Triangles

Similar triangles have the same shape. The lengths of their corresponding sides have the same ratio—that is, they are proportional.

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Example 8. The triangles below are similar triangles. Find the unknown length x. 30 x

33

10 12

11Solution: The ratio of x to 12 is the same as the ratio of 30 to 10 or 33 to 11.

Solve either proportion:

30

12 10

x

10 360 x

36x

Answer: 72

The lengths of the figures are proportional. Find x.

60 x

66

20 24

22


30 33 and

12 10 12 11

x x We get the proportions:

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In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers.

A positive number is written in scientific notation if it is written as a product of a number a, where 1 a < 10, and an integer power r of 10.

a 10r

Objective D: Scientific Notation

To Write a Number in Scientific Notation1. Move the decimal point in the original number to the left or right, so

that the new number has a value between 1 and 10.2. Count the number of decimal places the decimal point is moved in

Step 1. If the original number is 10 or greater, the count is positive.If the original number is less than 1, the count is negative.

3. Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.

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Example 9. Write each of the following in scientific notation.

a) 4 7 0 0 Move the decimal 3 places to the left, so that the new number has a value between 1 and 10.

Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3.

Answer: 4700 = 4.7 103

b) 0 . 0 0 0 4 7 Move the decimal 4 places to the right, so that the new number has a value between 1 and 10.

Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4.

Answer: 0.00047 = 4.7 10-4

5

8

Answers:

a) 5.62 10

b) 9.24 10


Write each of the following in scientific notation.

a) 0.0000562

b) 924,000,000

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To Write a Scientific Notation Number in Standard FormMove the decimal point the same number of spaces as the exponent on 10.If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

Example 10. Write each of the following in standard notation.

a) 5.2738 103 Since the exponent is a positive 3, we move the decimal 3 places to the right.

b) 6.45 10-5 Since the exponent is a negative 5, we move the decimal 5 places to the left.

Answer: 5.2738 103 = 5273.8

Answer: 00006.45 10-5 = 0.0000645

Your Turn Problem #10Write each of the following in scientific notation.

3

3

a) 2.4 10

b) 7.12 10

a) 2,400

b) 0.00712

Answers:The End.B.R.6-16-08

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7.2 Measurement – Section 2 1 Objective A: To Convert between Celsius and Fahrenheit using the formulas There are two temperature scales: Fahrenheit, which