Welcome to Interactive Chalkboard Pre-Algebra Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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Contents Lesson 9-1Squares and Square Roots Lesson 9-2The Real Number System Lesson 9-3Angles Lesson 9-4Triangles Lesson 9-5The Pythagorean Theorem Lesson 9-6The Distance and Midpoint Formulas Lesson 9-7Similar Triangles and Indirect Measurement Lesson 9-8Sine, Cosine, and Tangent Ratios

Lesson 1 Contents Example 1Find Square Roots Example 2Calculate Square Roots Example 3Estimate Square Roots Example 4Use Square Roots to Solve a Problem

Example 1-1a Find. indicates the positive square root of 64. Since Answer: 8

Example 1-1a Find. indicates the negative square root of 121. Since Answer: –11

Example 1-1a Since Answer: 2 and –2 Find. indicates both square roots of 4.

Example 1-1b Find each square root. a. b. c. Answer: 4 and –4 Answer: –12 Answer: 5

Example 1-2a Use a calculator to find to the nearest tenth. ENTER 2nd [] Use a calculator. Round to the nearest tenth. Check Since, the answer is reasonable. Answer: 4.8

Use a calculator to find to the nearest tenth. Example 1-2a ENTER 2nd [] Use a calculator. Round to the nearest tenth. Answer: –6.8 Check Since, the answer is reasonable.

Use a calculator to find each square root to the nearest tenth. a. b. Example 1-2b Answer: –6.2 Answer: 8.4

Example 1-3a Estimate to the nearest whole number. Find the two perfect squares closest to 22. To do this, list some perfect squares. 1, 4, 9, 16, 25, … 16 and 25 are closest to 22.

Example 1-3a and. 4<<54<<5 16<22<2522 is between 16 and 25. is betweenand. < Since 22 is closer to 25 than 16, the best whole number estimate for is 5. Answer: 5

Estimate to the nearest whole number. Example 1-3a Find the two perfect squares closest to 319. To do this, list some perfect squares...., 225, 256, 289, 324, … 289 and 324 are closest to 319.

Example 1-3a –324<–319<–289–319 is between –324 and –289. < is between and. –18<<–17 and.

Example 1-3a Answer: –18 Since –319 is closer to –324 than –289, the best whole number estimate for is –18. Check

Example 1-3b Estimate each square root to the nearest whole number. a. b. Answer: –12 Answer: 7

Example 1-4a Landmarks The observation deck at the Seattle Space Needle is 520 feet above the ground. On a clear day, about how far could a tourist on the deck see? Round to the nearest tenth. Use the formula where D is the distance in miles and A is the altitude, or height, in feet. Write the formula. Replace A with 520. Evaluate the square root first.

Example 1-4a Multiply. Answer:On a clear day, a tourist could see about 27.8 miles.

Example 1-4b Skyscraper A skyscraper stands 378 feet high. On a clear day, about how far could an individual standing on the roof of the skyscraper see? Round to the nearest tenth. Answer:On a clear day, an individual could see about 23.7 miles.

End of Lesson 1

Lesson 2 Contents Example 1Classify Real Numbers Example 2Compare Real Numbers on a Number Line Example 3Solve Equations

Example 2-1a Name all of the sets of numbers to which the real number 17 belongs. Answer:This number is a natural number, a whole number, an integer, and a rational number.

Example 2-1a Name all of the sets of numbers to which the real number belongs. Answer:Since, this number is an integer and a rational number.

Example 2-1a Answer:Since, this number is a natural number, a whole number, an integer, and a rational number. Name all of the sets of numbers to which the real number belongs.

Example 2-1a Name all of the sets of numbers to which the real number belongs. Answer:This repeating decimal is a rational number because it is equivalent to.

Example 2-1a Name all of the sets of numbers to which the real number belongs. Answer:It is not the square root of a perfect square so it is irrational.

Example 2-1b Name all of the sets of numbers to which each real number belongs. a. 31 b. c d. e. Answer:natural number, whole number, integer, rational number Answer:integer, rational number Answer:rational number Answer:natural number, whole number, integer, rational number Answer:irrational number

Example 2-2a Express each number as a decimal. Then graph the numbers. Replace  with, or = to make a true statement.

Example 2-2a Since is to the left of Answer:

Example 2-2a Express each number as a decimal. Then compare the decimals. Order from least to greatest.

Example 2-2a Answer: From least to greatest, the order is

Example 2-2b a.Replace  with, or = to make a true statement. b.Orderfrom least to greatest. Answer: > Answer:

Example 2-3a Solve. Round to the nearest tenth, if necessary. Answer:The solutions are 13 and –13. Write the equation. Take the square root of each side. Find the positive and negative square root.

Example 2-3a Answer:The solutions are 7.1 and –7.1. Solve. Round to the nearest tenth, if necessary. Write the equation. Take the square root of each side. Find the positive and negative square root. Use a calculator.

Solve each equation. Round to the nearest tenth, if necessary. a. b. Example 2-3b Answer: 9 and –9 Answer: 4.9 and –4.9

End of Lesson 2

Lesson 3 Contents Example 1Measure Angles Example 2Draw Angles Example 3Classify Angles Example 4Use Angles to Solve a Problem

Step 1 Place the center point of the protractor’s base on vertex S. Align the straight edge with side so that the marker for 0° is on the ray. Example 3-1a Use a protractor to measure  RSW.

Example 3-1a Step 2 Use the scale that begins with 0° at. Read where the other side of the angle,, crosses this scale. Use a protractor to measure  RSW. 42°

Example 3-1a Use a protractor to measure  RSW. 42° Answer:The measure of angle RSW is 42°. Using symbols,

Example 3-1a Find the measurements of  GUM,  SUM, and  BUG. Answer: is at 0° on the right. 120°

Example 3-1a Find the measurements of  GUM,  SUM, and  BUG. Answer: is at 0° on the right. 32°

Example 3-1a Find the measurements of  GUM,  SUM, and  BUG. Answer: is at 0° on the left. 60°

Example 3-1b a. Use a protractor to measure  ABC. Answer: 75°

Example 3-1b b. Find the measures of  FDE,  GDE, and  HDG. Answer:  FDE = 37°,  GDE = 118°,  HDG = 62°

Example 3-2a Draw  R having a measurement of 145°. Step 1Draw a ray with endpoint R. R

Example 3-2a Draw  R having a measurement of 145°. Step 2Place the center point of the protractor on R. Align the mark labeled 0 with the ray. R

Example 3-2a Draw  R having a measurement of 145°. 145° Step 3Use the scale that begins with 0. Locate the mark labeled 145. Then draw the other side of the angle. R

Example 3-2a Answer: 145° R

Example 3-2b Draw  M having a measurement of 47°. Answer:

Example 3-3a Classify the angle as acute, obtuse, right, or straight. m  KLM < 90. Answer:  KLM is acute.

Example 3-3a Classify the angle as acute, obtuse, right, or straight. m  NPQ = 180. Answer:  NPQ is straight.

Example 3-3a Classify the angle as acute, obtuse, right, or straight. m  RST > 90. Answer:  RST is obtuse.

Example 3-3b Classify each angle as acute, obtuse, right, or straight. a. b. Answer:right Answer:obtuse

Example 3-3b Classify each angle as acute, obtuse, right, or straight. c. Answer:straight

Example 3-4a The diagram shows the angle between the back of a chair and the seat of the chair. Classify this angle. Answer:Since 95° is greater than 90°, the angle is obtuse.

Example 3-4b The diagram shows the angle between the bed of the truck and the frame of the truck. Classify this angle. Answer:The angle is acute.

End of Lesson 3

Lesson 4 Contents Example 1Find Angle Measures Example 2Use Ratios to Find Angle Measures Example 3Classify Triangles

Example 4-1a Find the value of x in  DEF. The sum of the measures is 180. Replace m  D with 100 and m  E with 33. Simplify.

Example 4-1a Subtract 133 from each side. Answer:The measure of  F is 47°.

Example 4-1b Find the value of x in  MNO. Answer:The measure of  N is 57°.

Example 4-2a Algebra The measures of the angles of a certain triangle are in the ratio 2:3:5. What are the measures of the angles? WordsThe measures of the angles are in the ratio 2:3:5. VariablesLet 2x represent the measure of one angle, 3x the measure of a second angle, and 5x the measure of the third angle. Equation The sum of the measures is 180.

Example 4-2a Since Combine like terms. Divide each side by 10. Simplify. Answer:The measures of the angles are 36°, 54°, and 90°.

Example 4-2a Check So, the answer is correct.

Example 4-2b Algebra The measures of the angles of a certain triangle are in the ratio 3:5:7. What are the measures of the angles? Answer:The measures of the angles are 36°, 60°, and 84°.

Example 4-3a Classify the triangle by its angles and by its sides. AnglesAll angles are acute. SidesAll sides are congruent. Answer:The triangle is an acute equilateral triangle.

Example 4-3a Classify the triangle by its angles and by its sides. AnglesThe triangle has a right angle. SidesThe triangle has no congruent sides. Answer:The triangle is a right scalene triangle.

Example 4-3b Classify each triangle by its angles and by its sides. a. b. Answer:obtuse scalene Answer:acute equilateral

End of Lesson 4

Lesson 5 Contents Example 1Find the Length of the Hypotenuse Example 2Solve a Right Triangle Example 3Use the Pythagorean Theorem Example 4Identify a Right Triangle

Example 5-1a Find the length of the hypotenuse of the right triangle. Pythagorean Theorem Replace a with 21 and b with 20. Evaluate 21 2 and Add 441 and 400.

Example 5-1b Take the square root of each side. Answer:The length of the hypotenuse is 29 feet.

Example 5-1c Find the length of the hypotenuse of the right triangle. Answer:The length of the hypotenuse is 5 meters.

Example 5-2a Find the length of the leg of the right triangle. Pythagorean Theorem Replace c with 11 and a with 8. Evaluate 11 2 and 8 2.

Example 5-2b Subtract 64 from each side. Simplify. Take the square root of each side. ENTER 2nd [] Answer:The length of the leg is about 7.5 meters.

Example 5-2c Find the length of the leg of the right triangle. Answer:The length of the leg is about 12.7 inches.

Example 5-3a Multiple-Choice Test Item A building is 10 feet tall. A ladder is positioned against the building so that the base of the ladder is 3 feet from the building. How long is the ladder? A 12.4 feetB 10.4 feet C 10.0 feetD 14.9 feet Read the Test Item Make a drawing to illustrate the problem. The ladder, ground, and side of the house form a right triangle.

Example 5-3b Solve the Test Item Use the Pythagorean Theorem to find the length of the ladder. Pythagorean Theorem Replace a with 3 and b with 10. Evaluate 3 2 and Simplify.

Example 5-3c Take the square root of each side. Round to the nearest tenth. The ladder is about 10.4 feet tall. Answer:The answer is B.

Example 5-3d Multiple-Choice Test Item An 18-foot ladder is placed against a building which is 14 feet tall. About how far is the base of the ladder from the building? A 11.6 feetB 10.9 feet C 11.3 feetD 11.1 feet Answer:The answer is C.

Example 5-4a The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 48 ft, 60 ft, 78 ft The triangle is not a right triangle. Answer:no Pythagorean Theorem Replace a with 48, b with 60, and c with 78. Evaluate. Simplify.

Example 5-4b The measures of three sides of a triangle are given. Determine whether the triangle is a right triangle. 24 cm, 70 cm, 74 cm The triangle is a right triangle. Answer:yes Pythagorean Theorem Replace a with 24, b with 70, and c with 74. Evaluate. Simplify.

Example 5-4c The measures of three sides of a triangle are given. Determine whether each triangle is a right triangle. a.42 in., 61 in., 84 in. b.16 m, 30 m, 34 m Answer:no Answer:yes

End of Lesson 5

Lesson 6 Contents Example 1Use the Distance Formula Example 2Use the Distance Formula to Solve a Problem Example 3Use the Midpoint Formula

Example 6-1a Find the distance between M(8, 4) and N(–6, –2). Round to the nearest tenth, if necessary. Use the Distance Formula. Distance Formula Simplify.

Example 6-1b Evaluate (–14) 2 and (–6) 2. Add 196 and 36. Take the square root. Answer:The distance between points M and N is about 15.2 units.

Example 6-1c Find the distance between A(–4, 5) and B(3, –9). Round to the nearest tenth, if necessary. Answer:The distance between points A and B is about 15.7 units.

Example 6-2a Geometry Find the perimeter of  XYZ to the nearest tenth. First, use the Distance Formula to find the length of each side of the triangle.

Example 6-2b Distance Formula Simplify. Evaluate powers. Simplify.

Example 6-2c Distance Formula Simplify. Evaluate powers. Simplify.

Example 6-2d Distance Formula Simplify. Evaluate powers. Simplify.

Example 6-2e Then add the lengths of the sides to find the perimeter. Answer:The perimeter is about 15.8 units.

Example 6-2f Geometry Find the perimeter of  ABC to the nearest tenth. Answer:The perimeter is about 21.3 units.

Example 6-3a Find the coordinates of the midpoint of

Example 6-3b Substitution Simplify. Answer:The coordinates of the midpoint of are (3, 3). Midpoint Formula

Example 6-3c Find the coordinates of the midpoint of Answer:The coordinates of the midpoint of are (1, –1).

End of Lesson 6

Lesson 7 Contents Example 1Find Measures of Similar Triangles Example 2Use Indirect Measurement Example 3Use Shadow Reckoning

Example 7-1a If  RUN ~  CAB, what is the value of x ? The corresponding sides are proportional. Write a proportion.

Example 7-1b Replace UR with 4, AC with 8, UN with 10, and AB with x. Find the cross products. Simplify. Mentally divide each side by 4. Answer:The value of x is 20.

Example 7-1c If  ABC ~  DEF, what is the value of x ? Answer:The value of x is 3.

Example 7-2a Maps A surveyor wants to find the distance RS across the lake. He constructs  PQT similar to  PRS and measures the distances as shown. What is the distance across the lake?

Example 7-2b Substitution Find the cross products. Simplify. Divide each side by 25. Answer:The distance across the lake is 28.8 meters. Write a proportion.

Example 7-2c Maps In the figure,  MNO is similar to  OPQ. Find the distance across the park. Answer:The distance across the park is 4.8 miles.

Example 7-3a Landmarks Suppose the John Hancock Center in Chicago, Illinois, casts a foot shadow at the same time a nearby tourist casts a 1.5-foot shadow. If the tourist is 6 feet tall, how tall is the John Hancock Center? ExploreYou know the lengths of the shadows and the height of the tourist. You need to find the height of the John Hancock Center. Plan Write and solve a proportion. Solve tourist’s shadow building’s shadow tourist’s height building’s height

Example 7-3b Find the cross products. Multiply. Divide each side by 1.5. Answer:The height of the John Hancock Center is 1030 feet.

Example 7-3c Building A man standing near a building casts a 2.5-foot shadow at the same time the building casts a 200-foot shadow. If the man is 6 feet tall, how tall is the building? Answer:The height of the building is 480 feet.

End of Lesson 7

Lesson 8 Contents Example 1Find Trigonometric Ratios Example 2Use a Calculator to Find Trigonometric Ratios Example 3Use Trigonometric Ratios Example 4Use Trigonometric Ratios to Solve a Problem

Example 8-1a Find sin A, cos A, and tan A. Answer:

Example 8-1b Find sin A, cos A, and tan A. Answer:

Example 8-1c Find sin A, cos A, and tan A. Answer:

Example 8-1d Find sin B, cos B, and tan B. Answer:sin B = 0.8 ; cos B = 0.6 ; tan B =

Example 8-2a Find the value of sin 19° to the nearest ten thousandth. ENTER SIN Answer:sin 19° is about

Example 8-2b Find the value of cos 51° to the nearest ten thousandth. ENTER COS Answer:cos 51° is about

Example 8-2c Find the value of tan 24° to the nearest ten thousandth. ENTER TAN Answer:tan 24° is about

Example 8-2d Find each value to the nearest ten thousandth. a.sin 63° b.cos 14° c.tan 41° Answer: Answer: Answer:

Example 8-3a Find the missing measure. Round to the nearest tenth. The measures of an acute angle and the side adjacent to it are known. You need to find the measure of the hypotenuse. Use the cosine ratio. Write the cosine ratio.

Example 8-3b Substitution Multiply each side by x. Simplify. Divide each side by cos 71°.

Example 8-3c ENTER COS  Simplify. Answer:The measure of the hypotenuse is about 36.9 units.

Example 8-3d Find the missing measure. Round to the nearest tenth. Answer:The measure of the missing side is about 21.4 units.

Example 8-4a Architecture A tourist visiting the Petronas Towers in Kuala Lumpur, Malaysia, stands 261 feet away from their base. She looks at the top at an angle of 80° with the ground. How tall are the Towers? Use the tangent ratio. Write the tangent ratio.

Example 8-4b ENTER TAN X Substitution Multiply each side by 261. Simplify. Answer:The height of the Towers is about feet.

Example 8-4c Architecture Jenna stands 142 feet away from the base of a building. She looks at the top at an angle of 62° with the ground. How tall is the building? Answer:The building is about feet tall.

End of Lesson 8

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