chapter outline 8 measurement and geometry slide 2 copyright (c) the mcgraw-hill companies, inc....
TRANSCRIPT
CHAPTER OUTLINE
8Measurement and Geometry
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8.1 U.S. Customary Units of Measurement8.2 Metric Units of Measurement8.3 Converting Between U.S. Customary and Metric
Units8.4 Medical Applications Involving Measurement8.5 Lines and Angles8.6 Triangles and the Pythagorean Theorem
CHAPTER OUTLINE
8Measurement and Geometry
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8.7 Perimeter, Circumference, and Area8.8 Volume and Surface Area
Section
Objectives
8.1 U.S. Customary Units of Measurement
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1. U.S. Customary Units2. U.S. Customary Units of Length3. Units of Time4. U.S. Customary Units of Weight5. U.S. Customary Units of Capacity
Section 8.1 U.S. Customary Units of Measurement
1. U.S. Customary Units
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To measure an object means to assign it a number and a unit of measure.
Section 8.1 U.S. Customary Units of Measurement
1. U.S. Customary Units
(continued)
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Summary of U.S. Customary Units of Length, Time,Weight, and Capacity
Section 8.1 U.S. Customary Units of Measurement
1. U.S. Customary Units
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Section 8.1 U.S. Customary Units of Measurement
2. U.S. Customary Units of Length
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A conversion factor is a ratio of equivalentmeasures.
In a unit ratio, the quotient is 1 because we are dividing measurements of equal length. To convert from one unitof measure to another, we can multiply by a unit ratio.
PROCEDURE Choosing a Unit Ratio as a Conversion Factor
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In a unit ratio,• The unit of measure in the numerator should be the new unit you want to convert to.• The unit of measure in the denominator should be the original unit you want to convert from.
Example 1 Converting Units of Lengthby Using Unit Ratios
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Convert the units of length.
ExampleSolution:
1 Converting Units of Lengthby Using Unit Ratios
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a. From the table, we have 1 yd = 3 ft.
Notice that the original units of ft reduce or “cancel” in much the same way as simplifying fractions. The unit yd remains in the final answer.
ExampleSolution:
1 Converting Units of Lengthby Using Unit Ratios
(continued)
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b. From the table, we have 1 mi = 1760 yd.
ExampleSolution:
1 Converting Units of Lengthby Using Unit Ratios
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c. From the table we have 1 mi = 5280 ft.
Example 2 Making Multiple Conversions of Length
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ExampleSolution:
2 Making Multiple Conversions of Length
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a. To convert miles to inches, we use two conversion factors. The first unit ratio converts miles to feet. The second unit ratio converts feet to inches.
ExampleSolution:
2 Making Multiple Conversions of Length
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Example 3 Adding and Subtracting Mixed Units of Measurement
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ExampleSolution:
3 Adding and Subtracting Mixed Units of Measurement
(continued)
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ExampleSolution:
3 Adding and Subtracting Mixed Units of Measurement
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Example 5 Converting Units of Time
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After running a marathon, Dave crossed the finish line and noticed that the race clock read 2:20:30. Convert this time to minutes.
ExampleSolution:
5 Converting Units of Time
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The notation 2:20:30 means 2 hr 20 min 30 sec. We must convert 2 hr to minutes and 30 sec to minutes. Then we add the total number of minutes.
The total number of minutes is 120 min + 20 min + 0.5 min. Dave finished the race in 140.5 min.
Section 8.1 U.S. Customary Units of Measurement
4. U.S. Customary Units of Weight
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Measurements of weight record the force of an object subject to gravity.
Example 6 Converting Units of Weight
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a. The average weight of an adult male African elephant is 12,400 lb. Convert this value to tons.
b. Convert the weight of a 7-lb 3-oz baby to ounces.
ExampleSolution:
6 Converting Units of Weight
(continued)
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a. Recall that 1 ton = 2000 lb.
An adult male African elephant weighs 6.2 tons.
ExampleSolution:
6 Converting Units of Weight
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Example 7 Applying U.S. Customary Units of Weight
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Jessica lifts four boxes of books. The boxes have the following weights: 16 lb 4 oz, 18 lb 8 oz, 12 lb 5 oz, and 22 lb 9 oz. How much weight did she lift altogether?
ExampleSolution:
7 Applying U.S. Customary Units of Weight
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Section 8.1 U.S. Customary Units of Measurement
5. U.S. Customary Units of Capacity
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Capacity is the volume or amount that a container can hold. The U.S. Customary units of capacity are fluid ounces (fl oz), cup (c), pint (pt), quart (qt), and gallon (gal).
Example 8 Converting Units of Capacity
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Convert the units of capacity.
ExampleSolution:
8 Converting Units of Capacity
(continued)
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ExampleSolution:
8 Converting Units of Capacity
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Example
A recipe calls for c of chicken broth. A can ofchicken broth holds 14.5 fl oz. Is there enough chicken broth in the can for the recipe?
9 Applying Units of Capacity
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ExampleSolution:
9 Applying Units of Capacity
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We need to convert each measurement to the sameunit of measure for comparison.
The recipe calls for c or 14 fl oz of chicken broth. The can of chicken broth holds 14.5 fl oz, which is enough.
Section
Objectives
8.2 Metric Units of Measurement
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1. Introduction to the Metric System2. Metric Units of Length3. Metric Units of Mass4. Metric Units of Capacity5. Summary of Metric Conversions
Section 8.2 Metric Units of Measurement
1. Introduction to the Metric System
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The metric system, a simple, decimal-based system of units, is the predominant system of measurement used in science. The simplicity of the metric system is a result of having one basic unit of measure for each type of quantity (length, mass, and capacity). The base units are the meter for length, the gram for mass, and the liter for capacity. Other units of length, mass, and capacity in the metric system are products of the base unit and a power of 10.
Section 8.2 Metric Units of Measurement
2. Metric Units of Length
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The meter (m) is the basic unit of length in the metric system. A meter is slightly longer than a yard.
Section 8.2 Metric Units of Measurement
2. Metric Units of Length
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Metric Units of Length and Their Equivalents
Section 8.2 Metric Units of Measurement
2. Metric Units of Length
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Example 1 Measuring Distances in Metric Units
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Approximate the distance in centimeters and in millimeters.
ExampleSolution:
1 Measuring Distances in Metric Units
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The numbered lines on the ruler are units of centimeters. Each centimeter is divided into 10 mm. We see that the width of the penny is not quite 2 cm. We can approximate this distance as 1.8 cm or equivalently 18 mm.
Example 2 Converting Metric Units of Length
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ExampleSolution:
2 Converting Metric Units of Length
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From the table, 1 km = 1000 m.
Section 8.2 Metric Units of Measurement
2. Metric Units of Length
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The place positions in our numbering system are based on powers of 10. For this reason, when we multiply a number by 10, 100, or 1000, we move the decimal point 1, 2, or 3 places, respectively, to the right. Similarly, when we multiply by 0.1, 0.01, or 0.001, we move the decimal point to the left 1, 2, or 3 places, respectively. Since the metric system is also based on powers of 10, we can convertbetween two metric units of length by moving the decimal point.
Section 8.2 Metric Units of Measurement
2. Metric Units of Length
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PROCEDURE Using the Prefix Line to Convert Metric Units
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Step 1 To use the prefix line, begin at the point on the line corresponding to the original unit you are given.
Step 2 Then count the number of positions you need to move to reach the new unit of measurement.
Step 3 Move the decimal point in the original measured value the same direction and same number of places as on the prefix line.
Step 4 Replace the original unit with the new unit of measure.
Example 3 Using the Prefix Line to ConvertMetric Units of Length
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ExampleSolution:
3 Using the Prefix Line to ConvertMetric Units of Length
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Section 8.2 Metric Units of Measurement
3. Metric Units of Mass
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The fundamental unit of mass in the metric system is the gram (g).
Section 8.2 Metric Units of Measurement
3. Metric Units of Mass
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Metric Units of Mass and Their Equivalents
Section 8.2 Metric Units of Measurement
3. Metric Units of Mass
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Example 4 Converting Metric Units of Mass
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ExampleSolution:
4 Converting Metric Units of Mass
(continued)
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ExampleSolution:
4 Converting Metric Units of Mass
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Section 8.2 Metric Units of Measurement
4. Metric Units of Capacity
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The basic unit of capacity in the metric system is the liter (L).
Section 8.2 Metric Units of Measurement
4. Metric Units of Capacity
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Metric Units of Capacity and Their Equivalents
Section 8.2 Metric Units of Measurement
4. Metric Units of Capacity
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Section 8.2 Metric Units of Measurement
4. Metric Units of Capacity
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1 mL is also equivalent to a cubic centimeter (cc or cm3). The unit cc is often used to measure dosages of medicine.
Example 5 Converting Metric Units of Capacity
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ExampleSolution:
5 Converting Metric Units of Capacity
(continued)
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ExampleSolution:
5 Converting Metric Units of Capacity
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Section 8.2 Metric Units of Measurement
5. Summary of Metric Conversions
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Example 7 Converting Metric Units
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a. The distance between San Jose and Santa Clara is 26 km. Convert this to meters.
b. A bottle of canola oil holds 946 mL. Convert this to liters.
c. The mass of a bag of rice is 90,700 cg. Convert this to grams.
d. A dose of an antiviral medicine is 0.5 cc. Convert this to milliliters.
ExampleSolution:
7 Converting Metric Units
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Section
Objectives
8.3 Converting Between U.S. Customary and Metric Units
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1. Summary of U.S. Customary and Metric Unit Equivalents
2. Converting U.S. Customary and Metric Units3. Applications4. Units of Temperature
Example 1 Converting Metric Units to U.S. Customary Units
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ExampleSolution:
1 Converting Metric Units to U.S. Customary Units
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Section 8.3 Converting Between U.S. Customary and Metric Units
1. Summary of U.S. Customary and Metric Unit Equivalents
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Example 2 Converting Units of Length
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Fill in the blank. Round to two decimal places, if necessary.
ExampleSolution:
2 Converting Units of Length
(continued)
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ExampleSolution:
2 Converting Units of Length
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Example 3 Converting Units of Weight and Mass
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ExampleSolution:
3 Converting Units of Weight and Mass
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Example 4 Converting Units of Capacity
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ExampleSolution:
4 Converting Units of Capacity
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Example 5 Converting Units in an Application
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A 2-L bottle of soda sells for $2.19. A 32-oz bottle of soda sells for $1.59. Compare the price per quart of each bottle to determine the better buy.
ExampleSolution:
5 Converting Units in an Application
(continued)
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Note that 1 qt = 2 pt = 4 c = 32 fl oz. So a 32-ozbottle of soda costs $1.59 per quart. Next, if we can convert 2 L to quarts, we can compute the unit cost per quart and compare the results.
ExampleSolution:
5 Converting Units in an Application
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Now find the cost per quart.
The cost for the 2-L bottle is $1.04 per quart, whereas the cost for 32 oz is $1.59 per quart. Therefore, the 2-L bottle is the better buy.
Section 8.3 Converting Between U.S. Customary and Metric Units
4. Units of Temperature
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Section 8.3 Converting Between U.S. Customary and Metric Units
4. Units of Temperature
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FORMULA Conversions for Temperature Scale
Example 7 Converting Units of Temperature
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ExampleSolution:
7 Converting Units of Temperature
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Example 8 Converting Units of Temperature
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ExampleSolution:
8 Converting Units of Temperature
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Section
Objectives
8.4 Medical Applications Involving Measurement
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1. Additional Metric Units of Mass2. Medical Applications
Section 8.4 Medical Applications Involving Measurement
1. Additional Metric Units of Mass
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Sometimes doctors prescribe medicines in very small amounts. In these cases, it is sometimes more convenient to use units of micrograms. The abbreviation for microgram is mcg or sometimes g.
Example 1 Converting Units of Micrograms
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a. Convert.b. A doctor gives a heart patient an initial dose of 200 mcg of nitroglycerin. How many milligrams is this?
ExampleSolution:
1 Converting Units of Micrograms
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Example 2 Applying Metric Units of Measure to Medicine
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A doctor orders the antibiotic oxacillin for a child. The dosage is 12.5 mg of the drug per kilogram of the child’s body mass. This dosage is given 4 times a day.a. How much of the drug should a 24-kg child get in one dose?b. How much of the drug would the child get if she were on a 10-day course of the antibiotic?
ExampleSolution:
2 Applying Metric Units of Measure to Medicine
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a. We need to multiply the unit rate of 12.5 mg per kilogram times the child’s body mass.
b. For a 10-day course, we need to multiply 300 g by the number of doses per day (4), and the total number of days (10).
Section
Objectives
8.5 Lines and Angles
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1. Basic Definitions2. Naming and Measuring Angles3. Complementary and Supplementary Angles4. Parallel and Perpendicular Lines
Section 8.5 Lines and Angles
1. Basic Definitions
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Example 1 Identifying Points, Lines, Line Segments, and Rays
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ExampleSolution:
1 Identifying Points, Lines, Line Segments, and Rays
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Section 8.5 Lines and Angles
2. Naming and Measuring Angles
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An angle is a geometric figure formed by two rays that share a common endpoint. The common endpoint is called the vertex of the angle.
Section 8.5 Lines and Angles
2. Naming and Measuring Angles
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The most common unit to measure an angle is the degree, denoted by .
Section 8.5 Lines and Angles
2. Naming and Measuring Angles
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Approximate the measure of an angle by using a tool called a protractor.
Section 8.5 Lines and Angles
2. Naming and Measuring Angles
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Example 2 Measuring Angles
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Read the protractor to determine the measure of each angle.
Example 2 Measuring Angles
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ExampleSolution:
2 Measuring Angles
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Section 8.5 Lines and Angles
2. Naming and Measuring Angles
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Section 8.5 Lines and Angles
3. Complementary and Supplementary Angles
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Example 3 Identifying Supplementary andComplementary Angles
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ExampleSolution:
3 Identifying Supplementary andComplementary Angles
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Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
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Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
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Angles that share a side are called adjacent angles.
Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
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If two lines intersect at a right angle, they are perpendicular lines.
Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
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Lines L1 and L2 are parallel lines. If a third line m intersects the two parallel lines, eight angles are formed.
Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
(continued)
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Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
(continued)
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Section 8.5 Lines and Angles
4. Parallel and Perpendicular Lines
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Example 1 Finding the Measure of Angles in a Diagram
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Assume that lines L1 and L2 are parallel. Find the measure of each angle, and explain how the angle is related to the given angle of 65
ExampleSolution:
1 Finding the Measure of Angles in a Diagram
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Section
Objectives
8.6 Triangles and the Pythagorean Theorem
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1. Triangles2. Square Roots3. Pythagorean Theorem
PROPERTY Angles of a Triangle
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Example 1 Finding the Measure of Angles Within a Triangle
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ExampleSolution:
1 Finding the Measure of Angles Within a Triangle
(continued)
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ExampleSolution:
1 Finding the Measure of Angles Within a Triangle
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Section 8.6 Triangles and the Pythagorean Theorem
1. Triangles
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Section 8.6 Triangles and the Pythagorean Theorem
1. Triangles
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TIP:
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Sometimes we use tick marks to denote segments of equal length. Similarly, we sometimes use a small arc to denote angles of equal measure.
Example 2 Evaluating Squares and Square Roots
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ExampleSolution:
2 Evaluating Squares and Square Roots
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Section 8.6 Triangles and the Pythagorean Theorem
3. Pythagorean Theorem
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PROPERTY Pythagorean Theorem
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For any right triangle,
Using the letters a, b, and c to represent the legs and hypotenuse, respectively, we have
Example 3 Finding the Length of the Hypotenuse of a Right Triangle
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Find the length of the hypotenuse of the right triangle.
ExampleSolution:
3 Finding the Length of the Hypotenuse of a Right Triangle
(continued)
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The lengths of the legs are given.
Label the triangle, using a, b, and c. It does not matter which leg is labeled a and which is labeled b.
Apply the Pythagorean theorem.
Substitute a = 6 and b = 8.
Simplify.
ExampleSolution:
3 Finding the Length of the Hypotenuse of a Right Triangle
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The solution to this equation is the positive number, c, that when squared equals 100.
Simplify the square root of 100.
The solution may be checked using the Pythagorean theorem.
Example 4 Finding the Length of a Leg in a Right Triangle
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Find the length of the unknown side of the right triangle.
ExampleSolution:
4 Finding the Length of a Leg in a Right Triangle
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The solution may be checked by using the Pythagorean theorem.
Example 5 Using the Pythagorean Theorem in an Application
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When Barb swam across a river, the current carried her 300 yd downstream from her starting point. If the river is 400 yd wide, how far did Barb swim?
ExampleSolution:
5 Using the Pythagorean Theorem in an Application
(continued)
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We first familiarize ourselves with the problem and draw a diagram. The distance Barb actually swims is the hypotenuse of the right triangle. Therefore, we label this distance c.
ExampleSolution:
5 Using the Pythagorean Theorem in an Application
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Section
Objectives
8.7 Perimeter, Circumference, and Area
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1. Quadrilaterals2. Perimeter and Circumference3. Area
Section 8.7 Perimeter, Circumference, and Area
1. Quadrilaterals
(continued)
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Recall that a polygon is a flat figure formed by line segments connected at their ends. A four-sided polygon is called a quadrilateral.
Section 8.7 Perimeter, Circumference, and Area
1. Quadrilaterals
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Section 8.7 Perimeter, Circumference, and Area
2. Perimeter and Circumference
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Recall that the perimeter of a polygon is the distance around the figure. Also recall that the “perimeter” of a circle is called the circumference.
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FORMULA Perimeter and Circumference
Example 1 Finding Perimeter and Circumference
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ExampleSolution:
1 Finding Perimeter and Circumference
(continued)
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ExampleSolution:
1 Finding Perimeter and Circumference
(continued)
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ExampleSolution:
1 Finding Perimeter and Circumference
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Section 8.7 Perimeter, Circumference, and Area
3. Area
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Recall that the area of a region is the number of square units that can be enclosed within the region.
(continued)
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FORMULA Area Formulas
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FORMULA Area Formulas
Example 2 Finding Area
(continued)
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Example 2 Finding Area
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ExampleSolution:
2 Finding Area
(continued)
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a. The field is in the shape of a parallelogram. The base is 0.6 km and the height is 1.8 km.
The field is 1.08 km2.
ExampleSolution:
2 Finding Area
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b. To find the area of the matting only, we can subtract the inner area from the outer 8-in. by 10-in. area. In each case, apply the formula, A = lw.
The matting is 32 in.2
Example 3 Finding Area
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ExampleSolution:
3 Finding Area
(continued)
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ExampleSolution:
3 Finding Area
(continued)
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ExampleSolution:
3 Finding Area
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Example 4 Computing the Area of a Circle
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ExampleSolution:
4 Computing the Area of a Circle
(continued)
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ExampleSolution:
4 Computing the Area of a Circle
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Example 5 Finding Area for a Landscaping Application
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Sod can be purchased in palettes for $225. If a palette contains 240 ft2 of sod, how much will it cost to cover the area in the figure?
ExampleSolution:
5 Finding Area for a Landscaping Application
(continued)
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To find the total cost, we need to know the total number of square feet. Then we can determine how many 240-ft2 palettes are required.
ExampleSolution:
5 Finding Area for a Landscaping Application
(continued)
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ExampleSolution:
5 Finding Area for a Landscaping Application
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To determine how many 240-ft2 palettes of sod are required, divide the total area by 240 ft2.
Number of palettes:
The total cost for 20 palettes is
The cost for the sod is $4500.
Section
Objectives
8.8 Volume and Surface Area
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1. Volume2. Surface Area
Section 8.8 Volume and Surface Area
1. Volume
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Volume is another word for capacity.
Volume can be measured in cubic units.
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FORMULA
Notice that the volume formulas for these three figures are given by the product of the area of the base and the height of the figure:
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FORMULA
Example 1 Finding Volume
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Find the volume. Round to the nearest whole unit.
ExampleSolution:
1 Finding Volume
(continued)
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ExampleSolution:
1 Finding Volume
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We can visualize the volume by “layering” cubes that are each 1 in. high. The number of cubes in each layer is equal to 4 3 = 12. Each layer has 12 cubes, and there are 5 layers. Thus, the total number of cubes is 12 5 = 60 for a volume of 60 in3.
Example 2 Finding the Volume of a Cylinder
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ExampleSolution:
2 Finding the Volume of a Cylinder
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Example 3 Finding the Volume of a Sphere
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ExampleSolution:
3 Finding the Volume of a Sphere
(continued)
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ExampleSolution:
3 Finding the Volume of a Sphere
(continued)
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Example 4 Finding the Volume of a Cone
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ExampleSolution:
4 Finding the Volume of a Cone
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Section 8.8 Volume and Surface Area
2. Surface Area
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Surface area (often abbreviated SA) is the area of the surface of a three-dimensional object.
Section 8.8 Volume and Surface Area
2. Surface Area
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Example 5 Determining Surface Area
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Determine the surface area of the rectangular solid.
ExampleSolution:
5 Determining Surface Area
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Example 6 Determining Surface Area
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ExampleSolution:
6 Determining Surface Area
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