Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
9.1 Square Roots Goal: Find and approximate square roots of numbers. Vocabulary Square root: Perfect square: Radical expression: Example 1 Finding a Square Root Playground A community is building a playground on a square plot of land with an area of 625 square yards. What is the length of each side of the plot of land? 2 A 625 yd Solution Because length cannot be negative, it doesn't make sense to find the negative square root. The plot of land is a square with an area of 625 square yards, so the length of each side of the plot of land is the . 625 because 625. Answer: The length of each side of the plot of land is . Checkpoint Find the square roots of the number. 1. 9 Copyright © Holt McDougal. All rights reserved. 2. 49 3. 169 4. 196 Chapter 9 • Pre-Algebra Notetaking Guide 199 9.1 Square Roots Goal: Find and approximate square roots of numbers. Vocabulary Square A square root of a number n is a number m such that m2 n. root: Perfect A perfect square is a number that is the square of an square: integer. A radical expression is an expression that involves Radical expression: a radical sign. Example 1 Finding a Square Root Playground A community is building a playground on a square plot of land with an area of 625 square yards. What is the length of each side of the plot of land? 2 A 625 yd Solution Because length cannot be negative, it doesn't make sense to find the negative square root. The plot of land is a square with an area of 625 square yards, so the length of each side of the plot of land is the positive square root of 625 . 625 25 because 252 625. Answer: The length of each side of the plot of land is 25 yards . Checkpoint Find the square roots of the number. 1. 9 3, 3 Copyright © Holt McDougal. All rights reserved. 2. 49 7, 7 3. 169 13, 13 4. 196 14, 14 Chapter 9 • Pre-Algebra Notetaking Guide 199 Example 2 Approximating a Square Root Approximate 28 to the nearest integer. The perfect square closest to, but less than, 28 is square closest to, but greater than, 28 is and . So, 28 is between . This statement can be expressed by the compound inequality < 28 < . < 28 < Identify perfect squares closest to 28. < < 28 Take positive square root of each number. < < 28 Evaluate square root of each perfect square. Answer: Because 28 is closer to to . The perfect than to , 28 is closer . So, to the nearest integer, 28 ≈ than to . Checkpoint Approximate the square root to the nearest integer. 5. 46 6. 125 Example 3 7. 68.9 8. 87.5 Using a Calculator Use a calculator to approximate 636 . Round to the nearest tenth. Keystrokes 2nd 200 [] 636 Chapter 9 • Pre-Algebra Notetaking Guide Display ) Answer Copyright © Holt McDougal. All rights reserved. Example 2 Approximating a Square Root Approximate 28 to the nearest integer. The perfect square closest to, but less than, 28 is 25 . The perfect square closest to, but greater than, 28 is 36 . So, 28 is between 25 and 36 . This statement can be expressed by the compound inequality 25 < 28 < 36 . 25 < 28 < 36 Identify perfect squares closest to 28. 25 < 28 < 36 < 6 5 < 28 Take positive square root of each number. Evaluate square root of each perfect square. Answer: Because 28 is closer to 25 than to 36 , 28 is closer to 5 than to 6 . So, to the nearest integer, 28 ≈ 5 . Checkpoint Approximate the square root to the nearest integer. 5. 46 6. 125 11 7 Example 3 7. 68.9 8. 87.5 9 8 Using a Calculator Use a calculator to approximate 636 . Round to the nearest tenth. Keystrokes 2nd 200 [] 636 Chapter 9 • Pre-Algebra Notetaking Guide ) Display Answer 25.21904043 25.2 Copyright © Holt McDougal. All rights reserved. Checkpoint Use a calculator to approximate the square root. Round to the nearest tenth. 9. 6 10. 104 11. 819 12. 1874 Evaluating a Radical Expression Example 4 Evaluate 5 a2 b when a 6 and b 27. a2 b 5 Substitute for a and for b. Evaluate expression inside radical symbol. Evaluate square root. Multiply. Checkpoint Evaluate the expression when a 16 and b 9. 13. a b Copyright © Holt McDougal. All rights reserved. 14. b2 2a 15. 2ab Chapter 9 • Pre-Algebra Notetaking Guide 201 Checkpoint Use a calculator to approximate the square root. Round to the nearest tenth. 9. 6 10. 104 11. 819 10.2 28.6 2.4 Example 4 12. 1874 43.3 Evaluating a Radical Expression Evaluate 5 a2 b when a 6 and b 27. a2 b 5 5 62 27 59 Substitute for a and for b. Evaluate expression inside radical symbol. 5p3 Evaluate square root. 15 Multiply. Checkpoint Evaluate the expression when a 16 and b 9. 13. a b 5 Copyright © Holt McDougal. All rights reserved. 14. b2 2a 7 15. 2ab 24 Chapter 9 • Pre-Algebra Notetaking Guide 201 9.2 Simplifying Square Roots Goal: Simplify radical expressions. Vocabulary Simplest form: Product Property of Square Roots Algebra ab a p b , where a ≥ 0 and b ≥ 0 Numbers 9 p 7 9 p 7 37 Example 1 150 Factor using greatest perfect square factor. Product property of square roots Simplify. Example 2 18t 2 202 Simplifying a Radical Expression Simplifying a Variable Expression Factor using greatest perfect square factor. Product property of square roots Simplify. Commutative property Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9.2 Simplifying Square Roots Goal: Simplify radical expressions. Vocabulary A radical expression is in simplest form when (1) no factor of the expression under the radical sign has any perfect square factor other than 1, (2) there are no fractions under the radical sign, and (3) there are no radical signs in the denominator of any fraction. Simplest form: Product Property of Square Roots Algebra ab a p b , where a ≥ 0 and b ≥ 0 Numbers 9 p 7 9 p 7 37 Example 1 Simplifying a Radical Expression 150 25 p6 25 p 6 Product property of square roots 56 Simplify. Example 2 Simplifying a Variable Expression 18t 2 9 p 2 p t2 202 Factor using greatest perfect square factor. Factor using greatest perfect square factor. 9 p 2 p t2 Product property of square roots 3 p 2 pt Simplify. 3t 2 Commutative property Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Checkpoint Simplify the expression. 1. 75 2. 80 3. 24r 2 4. 56m2 Quotient Property of Square Roots Algebra a , where a ≥ 0 and b > 0 ab b Numbers 11 11 11 4 2 4 Example 3 17 25 Simplifying a Radical Expression Quotient property of square roots Simplify. Checkpoint Simplify the expression. 5. Copyright © Holt McDougal. All rights reserved. 5 9 6. 21 64 7. 16 z 49 8. 32y2 81 Chapter 9 • Pre-Algebra Notetaking Guide 203 Checkpoint Simplify the expression. 1. 75 2. 80 53 3. 24r 2 4. 56m2 2r 6 2m14 45 Quotient Property of Square Roots Algebra a , where a ≥ 0 and b > 0 ab b Numbers 11 11 11 4 2 4 Example 3 17 25 Simplifying a Radical Expression 17 25 17 5 Quotient property of square roots Simplify. Checkpoint Simplify the expression. 5. 5 9 5 3 Copyright © Holt McDougal. All rights reserved. 6. 21 64 21 8 7. 16 z 49 4z 7 8. 32y2 81 4y 2 9 Chapter 9 • Pre-Algebra Notetaking Guide 203 Example 4 Using Radical Expressions The expression to fall h feet. h gives the time (in seconds) it takes an object 16 a. Write the expression in simplest form. b. Find the length of time it takes for an object to fall 120 feet. Give your answer to the nearest tenth of a second. Solution a. h 16 Quotient property of square roots Simplify. Answer: In simplest form, b. h ≈ h 16 . Substitute 120 for h. Approximate using a calculator. Answer: The length of time it takes for an object to fall 120 feet is about . Checkpoint 9. Find the length of time it takes for an object to fall 155 feet. Give your answer to the nearest tenth of a second. 204 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Using Radical Expressions Example 4 The expression to fall h feet. h gives the time (in seconds) it takes an object 16 a. Write the expression in simplest form. b. Find the length of time it takes for an object to fall 120 feet. Give your answer to the nearest tenth of a second. Solution a. h 16 h 16 h 4 Quotient property of square roots Simplify. Answer: In simplest form, h 120 b. 4 h 16 h . 4 Substitute 120 for h. 4 ≈ 2.7 Approximate using a calculator. Answer: The length of time it takes for an object to fall 120 feet is about 2.7 seconds . Checkpoint 9. Find the length of time it takes for an object to fall 155 feet. Give your answer to the nearest tenth of a second. 3.1 sec 204 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Focus On Operations Use after Lesson 9.2 Performing Operations on Square Roots Goal: Perform operations on square roots. Example 1 Multiplying and Dividing Radical Expressions a. 12 · 2 12 ·2 · Product property of square roots Multiply. Product property of square roots Simplify. b. 18 10 The process of rewriting a fraction so that its denominator is a rational number is called rationalizing the denominator. Quotient property of square roots Simplify. Quotient property of square roots Simplify. · Rationalize denominator. Multiply and simplify. Example 2 Adding and Subtracting Radical Expressions a. 56 32 46 32 ) 32 ( 32 b. 93 12 93 · 93 93 ( ) Copyright © Holt McDougal. All rights reserved. · Commutative property Distributive property Simplify. Factor using perfect square factor. Product property of square roots Simplify. Distributive property Simplify. Chapter 9 • Pre-Algebra Notetaking Guide 205 Focus On Operations Use after Lesson 9.2 Performing Operations on Square Roots Goal: Perform operations on square roots. Example 1 Multiplying and Dividing Radical Expressions a. 12 · 2 12 ·2 24 4 · 6 26 18 b. 18 10 10 9 Quotient property of square roots Simplify. 5 9 5 3 5 5 3 · 5 5 5 3 5 The process of rewriting a fraction so that its denominator is a rational number is called rationalizing the denominator. Example 2 Product property of square roots Multiply. Product property of square roots Simplify. Quotient property of square roots Simplify. Rationalize denominator. Multiply and simplify. Adding and Subtracting Radical Expressions a. 56 32 46 56 46 32 ( 5 4 ) 6 32 Commutative property Distributive property 6 32 Simplify. b. 93 12 93 Factor using perfect square 4·3 factor. 93 4 · 3 Product property of square roots 93 23 Simplify. ( 9 2 ) 3 113 Copyright © Holt McDougal. All rights reserved. Distributive property Simplify. Chapter 9 • Pre-Algebra Notetaking Guide 205 Checkpoint Simplify the expression. 1. 12 · 6 2. 48 21 3. 58 62 4. 83 311 3 Example 3 Using Radical Expressions gives the radius (in Circular plate The expression 3.14 centimeters) of a circular plate with area A (in square centimeters). About how much longer is the radius of a circular plate with an area of 628 square centimeters than the radius of a circular plate with an area of 157 square centimeters? A 1. Find the radius when A 628. A 3.14 Substitute for A. Divide. · 2 Product property of square roots Simplify. 2. Find the radius when A 157. A 3.14 Substitute for A. Divide. · 2 Product property of square roots Simplify. 3. Find the difference of the lengths. 102 52 ( Answer about 206 Chapter 9 • Pre-Algebra Notetaking Guide ) ≈ Distributive property · 1.4 ≈ Simplify and approximate. centimeters Copyright © Holt McDougal. All rights reserved. Checkpoint Simplify the expression. 1. 12 · 6 2. 48 21 62 47 7 3. 58 62 4. 83 311 3 73 3 11 162 Example 3 Using Radical Expressions gives the radius (in Circular plate The expression 3.14 centimeters) of a circular plate with area A (in square centimeters). About how much longer is the radius of a circular plate with an area of 628 square centimeters than the radius of a circular plate with an area of 157 square centimeters? A 1. Find the radius when A 628. 628 A 3.14 3.14 Substitute 628 for A. 200 Divide. 100 · 2 Product property of square roots 102 Simplify. 2. Find the radius when A 157. A 3.14 3.14 157 Substitute 157 for A. 50 Divide. · 2 25 Product property of square roots 52 Simplify. 3. Find the difference of the lengths. 102 52 ( 10 5 ) 2 52 ≈ 5 · 1.4 ≈ 7 Distributive property Simplify and approximate. Answer about 7 centimeters 206 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9.3 The Pythagorean Theorem Goal: Use the Pythagorean theorem to solve problems. Vocabulary Hypotenuse: Legs: Pythagorean Theorem Words For any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Example 1 2 2 2 ft 24 ft a2 b2 c2 Pythagorean theorem c2 Substitute for a and for b. c2 Evaluate powers and add. c Take positive square root of each side. ≈c Simplify. 2 Answer: The length of the ramp is about Copyright © Holt McDougal. All rights reserved. b Finding the Length of a Hypotenuse A building’s access ramp has a horizontal distance of 24 feet and a vertical distance of 2 feet. Find the length of the ramp to the nearest tenth of a foot. 2 c a legs Algebra a b c 2 hypotenuse feet. Chapter 9 • Pre-Algebra Notetaking Guide 207 9.3 The Pythagorean Theorem Goal: Use the Pythagorean theorem to solve problems. Vocabulary Hypotenuse: In a right triangle, the hypotenuse is the side opposite the right angle. In a right triangle, the legs are the sides that form the right angle. Legs: Pythagorean Theorem Words For any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. 2 2 A building’s access ramp has a horizontal distance of 24 feet and a vertical distance of 2 feet. Find the length of the ramp to the nearest tenth of a foot. 2 b Finding the Length of a Hypotenuse Example 1 2 c a legs Algebra a b c 2 hypotenuse a2 b2 c2 Pythagorean theorem 24 c2 Substitute for a and for b. 580 c2 Evaluate powers and add. 2 580 c 24.1 ≈ c 2 ft 24 ft Take positive square root of each side. Simplify. Answer: The length of the ramp is about 24.1 feet. Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 207 Finding the Length of a Leg Example 2 Find the unknown length a in simplest form. a2 b2 c2 a2 2 a2 Pythagorean theorem 2 c 15 a Substitute. b 10 Evaluate powers. a2 Subtract from each side. a Take positive square root of each side. a Simplify. Answer: The unknown length a is units. Checkpoint Find the unknown length. Write your answer in simplest form. 1. 2. a6 c a c 13 3. a 11 c 20 b 12 b b8 Converse of the Pythagorean Theorem The Pythagorean theorem can be written in “if-then” form. Theorem: If a triangle is a right triangle, then a2 b2 c2. If you reverse the two parts of the statement, the new statement is called the converse of the Pythagorean theorem. Although not all converses of true statements are true, the converse of the Pythagorean theorem is true. 208 Converse: If a2 b2 c2, then the triangle is a right triangle. Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Finding the Length of a Leg Example 2 Find the unknown length a in simplest form. a2 b2 c2 a2 10 2 15 Pythagorean theorem 2 c 15 a Substitute. a2 100 225 b 10 Evaluate powers. a2 125 Subtract 100 from each side. a 125 Take positive square root of each side. a 55 Simplify. units. Answer: The unknown length a is 55 Checkpoint Find the unknown length. Write your answer in simplest form. 1. 2. c a6 a c 13 3. a 11 c 20 b 12 b b8 10 5 331 Converse of the Pythagorean Theorem The Pythagorean theorem can be written in “if-then” form. Theorem: If a triangle is a right triangle, then a2 b2 c2. If you reverse the two parts of the statement, the new statement is called the converse of the Pythagorean theorem. Although not all converses of true statements are true, the converse of the Pythagorean theorem is true. 208 Converse: If a2 b2 c2, then the triangle is a right triangle. Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Example 3 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a. a 8, b 9, c 12 Solution a. a2 b2 c2 2 2 b. a 7, b 24, c 25 b. 2 Answer: a2 b2 c2 2 2 2 Answer: Checkpoint Determine whether the triangle with the given side lengths is a right triangle. 4. a 12, b 9, c 15 Copyright © Holt McDougal. All rights reserved. 5. a 10, b 25, c 27 Chapter 9 • Pre-Algebra Notetaking Guide 209 Example 3 Identifying Right Triangles Determine whether the triangle with the given side lengths is a right triangle. a. a 8, b 9, c 12 Solution a. a2 b2 c2 8 2 9 2 12 b. 2 64 81 144 145 b. a 7, b 24, c 25 144 Answer: a2 b2 c2 7 2 24 2 25 2 49 576 625 625 625 Answer: Not a right triangle A right triangle Checkpoint Determine whether the triangle with the given side lengths is a right triangle. 4. a 12, b 9, c 15 yes Copyright © Holt McDougal. All rights reserved. 5. a 10, b 25, c 27 no Chapter 9 • Pre-Algebra Notetaking Guide 209 Focus On Geometry Use after Lesson 9.3 Angles of a Triangle Goal: Determine the sum of the measures of the angles of a triangle. Vocabulary Acute triangle 60° 45° 75° Right triangle 55° 35° Obtuse triangle 40° 105° Equiangular triangle 35° 60° 60° 60° Sum of the Measures of the Angles of a Triangle C The sum of the measures of the interior angles of a triangle is 180°. B A maA 1 maB 1 maC 5 180° Example 1 Finding an Angle Measure in a Triangle Find the value of x. X° 32° 54° 54° + + x° 5 +x 5 x 5 210 Chapter 9 • Pre-Algebra Notetaking Guide Sum of angle measures is 180°. Add. Subtract from each side. Copyright © Holt McDougal. All rights reserved. Focus On Geometry Use after Lesson 9.3 Angles of a Triangle Goal: Determine the sum of the measures of the angles of a triangle. Vocabulary Acute triangle 60° 3 acute angles 45° 75° Right triangle 55° 1 right angle 35° Obtuse triangle 40° 1 obtuse angle 105° Equiangular triangle 35° 60° 3 congruent angles 60° 60° Sum of the Measures of the Angles of a Triangle C The sum of the measures of the interior angles of a triangle is 180°. B A maA 1 maB 1 maC 5 180° Example 1 Finding an Angle Measure in a Triangle Find the value of x. X° 32° 54° 54° + 32° + x° 5 180° 86 + x 5 180 x 5 94 210 Chapter 9 • Pre-Algebra Notetaking Guide Sum of angle measures is 180°. Add. Subtract 86 from each side. Copyright © Holt McDougal. All rights reserved. Classifying a Triangle by its Angle Measures Example 2 Find the values of x, y, and z. Then classify all triangles in the diagram by their angle measures. B 64° 34° A x° 102° y° D z° C 1. Write and solve an equation to find the value of x. x° + 34° + = Sum of angle measures is 180°. x+ = Add. x= Subtract from each side. 2. Write and solve an equation to find the value of y. y° + = The measure of a straight angle is 180°. y= Subtract from each side. 3. Write and solve an equation to find the value of z. z° + 78° + = Sum of angle measures is 180°. z+ = Add. z= Subtract from each side. 4. TABD has , so it is a(n) triangle. TBCD has , so it is an triangle. The measure of aB is 34° + = , so TABC has and is a(n) triangle. Checkpoint Complete the following exercise. 1. Find the values of x, y, and z. Then classify all triangles in the diagram by their angle measures. B x° z° A Copyright © Holt McDougal. All rights reserved. 26° 120° y° D 60° C Chapter 9 • Pre-Algebra Notetaking Guide 211 Classifying a Triangle by its Angle Measures Example 2 Find the values of x, y, and z. Then classify all triangles in the diagram by their angle measures. B 64° 34° A x° 102° y° D z° C 1. Write and solve an equation to find the value of x. x° + 34° + 102° = 180° Sum of angle measures is 180°. Add. x + 136 = 180 Subtract 136 from each side. x = 44 2. Write and solve an equation to find the value of y. y° + 102° = 180° The measure of a straight angle is 180°. y = 78° Subtract 102 from each side. 3. Write and solve an equation to find the value of z. z° + 78° + 64° = 180° Sum of angle measures is 180°. Add. z + 142 = 180 Subtract 142 from each side. z = 38 4. T ABD has one obtuse angle , so it is a(n) obtuse triangle. T BCD has three acute angles , so it is an acute triangle. The measure of aB is 34° + 64° = 98° , so T ABC has one obtuse angle and is a(n) obtuse triangle. Checkpoint Complete the following exercise. 1. Find the values of x, y, and z. Then classify all triangles in the diagram by their angle measures. B x = 34, y = 60, z = 60; T ABD is obtuse, T BCD is equiangular, T ABC is obtuse x° z° A Copyright © Holt McDougal. All rights reserved. 26° 120° y° D 60° C Chapter 9 • Pre-Algebra Notetaking Guide 211 9.4 Real Numbers Goal: Compare and order real numbers. Vocabulary Irrational number: Real numbers: Classifying Real Numbers Example 1 Number Decimal Form 7 10 7 10 b. 2 9 2 9 c. 5 5 a. The square root of any whole number that is not a perfect square is irrational. Example 2 Decimal Type Type Comparing Real Numbers 6 5 Copy and complete 3 __ ? using <, >, or . Solution 6 5 Graph 3 and on a number line. 3 is to the 0 Answer: 3 212 Chapter 9 • Pre-Algebra Notetaking Guide 1 6 of . 5 2 6 5 Copyright © Holt McDougal. All rights reserved. 9.4 Real Numbers Goal: Compare and order real numbers. Vocabulary Irrational An irrational number is a number that cannot be number: written as a quotient of two integers. Real numbers consist of all rational and irrational Real numbers: numbers. Example 1 Classifying Real Numbers Number Decimal Form Type 7 10 7 0.7 10 Terminating Rational b. 2 9 2 0.222. . . 0.2 9 Repeating Rational c. 5 5 2.23606797. . . Nonterminating, Irrational a. The square root of any whole number that is not a perfect square is irrational. Decimal Type nonrepeating Example 2 Comparing Real Numbers 6 5 Copy and complete 3 __ ? using <, >, or . Solution 6 5 Graph 3 and on a number line. 6 5 6 3 is to the right of . 3 5 0 1 2 6 5 Answer: 3 > 212 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Ordering Real Numbers Example 3 6 5 Use a number line to order the numbers 2 , 2.2, 2, and from least to greatest. 22 Graph the numbers on a number line and read them from left to right. 3 2 1 0 1 2 3 Answer: From least to greatest, the numbers are . Checkpoint Tell whether the number is rational or irrational. 8 11 1. 2. 7 3. 49 4. 25 Copy and complete the statement using <, >, or . 3 2 5. __ ? 3 Copyright © Holt McDougal. All rights reserved. 6. 10 __ ? 3.5 Chapter 9 • Pre-Algebra Notetaking Guide 213 Ordering Real Numbers Example 3 6 5 Use a number line to order the numbers 2 , 2.2, 2, and from least to greatest. 22 Graph the numbers on a number line and read them from left to right. 6 2 2 2 2.2 3 2 1 0 1 5 2 2 3 Answer: From least to greatest, the numbers are 5 6 22 , 2.2, , and . 2 2 Checkpoint Tell whether the number is rational or irrational. 8 11 2. 7 1. rational irrational 3. 49 4. rational 25 irrational Copy and complete the statement using <, >, or . 3 2 5. __ ? 3 > Copyright © Holt McDougal. All rights reserved. 6. 10 __ ? 3.5 < Chapter 9 • Pre-Algebra Notetaking Guide 213 Checkpoint Use a number line to order the numbers from least to greatest. 27 5 7. 5.9, 35 , , 35 9 2 8. 21 , 4.6, , 25 Example 4 Using Irrational Numbers Speed After an accident, a police officer finds that the length of a car’s skid marks is 98 feet. The car’s speed s (in miles per hour) . and the length l (in feet) of the skid marks are related by s 27l Find the car’s speed to the nearest tenth of a mile per hour. Solution s 27l 27 p ≈ Write formula. Substitute value. Multiply. Approximate using a calculator. Answer: The car’s speed was about 214 Chapter 9 • Pre-Algebra Notetaking Guide miles per hour. Copyright © Holt McDougal. All rights reserved. Checkpoint Use a number line to order the numbers from least to greatest. 27 5 7. 5.9, 35 , , 35 27 , 5.9, 35 , 35 5 9 2 8. 21 , 4.6, , 25 9 2 4.6, 21 , , 25 Example 4 Using Irrational Numbers Speed After an accident, a police officer finds that the length of a car’s skid marks is 98 feet. The car’s speed s (in miles per hour) . and the length l (in feet) of the skid marks are related by s 27l Find the car’s speed to the nearest tenth of a mile per hour. Solution s 27l 27 p 98 2646 ≈ 51.4 Write formula. Substitute value. Multiply. Approximate using a calculator. Answer: The car’s speed was about 51.4 miles per hour. 214 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9.5 Distance and Midpoint Formulas Goal: Use the distance, midpoint, and slope formulas. The Distance Formula Words The distance between two points in a coordinate plane is equal to the square root of the sum of the horizontal change squared and the vertical change squared. 2 x2 x1 ) (y2 y1 )2 Algebra d ( Example 1 y (x2, y2) d (x1, y1) y2 y1 x2 x1 x O Finding the Distance Between Two Points Find the distance between the points M(2, 3) and N(1, 5). 2 d ( x2 x1 ) (y2 y1 )2 (1 ) (5 ) 2 2 ( ) Distance formula 2 2 Substitute. Subtract. Evaluate powers. Add. Answer: The distance between the points is units. Checkpoint Find the distance between the points. Write your answer in simplest form. 1. (2, 5), (3, 7) Copyright © Holt McDougal. All rights reserved. 2. (2, 2), (0, 4) Chapter 9 • Pre-Algebra Notetaking Guide 215 9.5 Distance and Midpoint Formulas Goal: Use the distance, midpoint, and slope formulas. The Distance Formula Words The distance between two points in a coordinate plane is equal to the square root of the sum of the horizontal change squared and the vertical change squared. 2 x2 x1 ) (y2 y1 )2 Algebra d ( Example 1 y (x2, y2) d (x1, y1) y2 y1 x2 x1 x O Finding the Distance Between Two Points Find the distance between the points M(2, 3) and N(1, 5). 2 d ( x2 x1 ) (y2 y1 )2 Distance formula (1 2 ) (5 3 ) 2 2 ( 1 ) 2 1 4 2 2 5 Substitute. Subtract. Evaluate powers. Add. units. Answer: The distance between the points is 5 Checkpoint Find the distance between the points. Write your answer in simplest form. 1. (2, 5), (3, 7) 13 Copyright © Holt McDougal. All rights reserved. 2. (2, 2), (0, 4) 210 Chapter 9 • Pre-Algebra Notetaking Guide 215 The Midpoint Formula Words The coordinates of the midpoint of a segment are the average of the endpoints’ x-coordinates and the average of the endpoints’ y-coordinates. y y1 y2 2 Example 2 M (x 1 A(x1, y1) x2 y1 y2 , 2 2 ) y1 x1 x2 y1 y2 , Algebra M 2 2 B(x2, y2) y2 O x1 x1 x2 x2 2 x Finding a Midpoint Find the midpoint M of the segment with endpoints (2, 5) and (6, 3). x x 2 y y 2 1 2 1 2 , M Midpoint formula 2 2 , Substitute values. Simplify. Checkpoint Find the midpoint of the segment with the given endpoints. 3. (2, 5), (3, 7) 216 Chapter 9 • Pre-Algebra Notetaking Guide 4. (3, 4), (7, 2) Copyright © Holt McDougal. All rights reserved. The Midpoint Formula Words The coordinates of the midpoint of a segment are the average of the endpoints’ x-coordinates and the average of the endpoints’ y-coordinates. x1 x2 y1 y2 , Algebra M 2 2 y B(x2, y2) y2 y1 y2 2 M (x 1 A(x1, y1) x2 y1 y2 , 2 2 ) y1 O x1 x1 x2 x2 2 x Finding a Midpoint Example 2 Find the midpoint M of the segment with endpoints (2, 5) and (6, 3). x x 2 y y 2 1 2 1 2 , M Midpoint formula 2 (6) 5 (3) , 2 2 (2, 1) Substitute values. Simplify. Checkpoint Find the midpoint of the segment with the given endpoints. 3. (2, 5), (3, 7) 12, 1 216 Chapter 9 • Pre-Algebra Notetaking Guide 4. (3, 4), (7, 2) (5, 1) Copyright © Holt McDougal. All rights reserved. Slope If points A(x1, y1) and B(x2, y2) do not lie on a vertical line, you can use coordinate notation to write a formula for the slope of the line through A and B. y y x2 x1 difference of y-coordinates difference of x-coordinates 2 1 slope Example 3 Finding Slope Find the slope of the line through (2, 1) and (1, 2). y y x2 x1 2 1 slope Slope formula Substitute values. Simplify. Checkpoint Find the slope of the line through the given points. 5. (2, 6), (3, 9) Copyright © Holt McDougal. All rights reserved. 6. (6, 2), (4, 7) Chapter 9 • Pre-Algebra Notetaking Guide 217 Slope If points A(x1, y1) and B(x2, y2) do not lie on a vertical line, you can use coordinate notation to write a formula for the slope of the line through A and B. y y x2 x1 difference of y-coordinates difference of x-coordinates 2 1 slope Example 3 Finding Slope Find the slope of the line through (2, 1) and (1, 2). y y x2 x1 2 1 slope Slope formula 2 (1) 1 2 Substitute values. 3 1 3 Simplify. Checkpoint Find the slope of the line through the given points. 5. (2, 6), (3, 9) 3 Copyright © Holt McDougal. All rights reserved. 6. (6, 2), (4, 7) 9 10 Chapter 9 • Pre-Algebra Notetaking Guide 217 9.6 Special Right Triangles Goal: Use special right triangles to solve problems. 45-45-90 Triangle Words In a 45-45-90 triangle, the length of the hypotenuse is the product of the length of a leg . and 2 45 a 2 a 45 a Algebra hypotenuse leg p 2 a2 Example 1 Using a 45-45-90 Triangle A 45-45-90 triangle used in mechanical drawing has 10-inch legs. Find the length of the hypotenuse to the nearest tenth of an inch. 45 10 in. 45 Solution 10 in. hypotenuse leg p 2 ≈ p 2 Rule for 45-45-90 triangle Substitute. Use a calculator. Answer: The length of the triangle’s hypotenuse is about inches. Checkpoint 1. Find the unknown length x. Write your answer in simplest form. 45 6 x 45 6 218 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9.6 Special Right Triangles Goal: Use special right triangles to solve problems. 45-45-90 Triangle Words In a 45-45-90 triangle, the length of the hypotenuse is the product of the length of a leg . and 2 45 a 2 a 45 a Algebra hypotenuse leg p 2 a2 Example 1 Using a 45-45-90 Triangle A 45-45-90 triangle used in mechanical drawing has 10-inch legs. Find the length of the hypotenuse to the nearest tenth of an inch. 45 10 in. 45 Solution 10 in. hypotenuse leg p 2 Rule for 45-45-90 triangle 10 p 2 Substitute. ≈ 14.1 Use a calculator. Answer: The length of the triangle’s hypotenuse is about 14.1 inches. Checkpoint 1. Find the unknown length x. Write your answer in simplest form. 45 6 x 45 62 218 Chapter 9 • Pre-Algebra Notetaking Guide 6 Copyright © Holt McDougal. All rights reserved. 30-60-90 Triangle In a 30-60-90 triangle, the shorter leg is opposite the 30 angle, and the longer leg is opposite the 60 angle. Words In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is the product of the length of the shorter . leg and 3 30 2a a 3 60 a Algebra hypotenuse 2 p shorter leg 2a longer leg shorter leg p 3 a3 Using a 30-60-90 Triangle Example 2 Find the length x of the hypotenuse and the length y of the longer leg of the triangle. 12 x 60 30 y The triangle is a 30-60-90 triangle. The length of the shorter leg is units. a. hypotenuse 2 p shorter leg x2p Answer: The length x of the hypotenuse is units. b. longer leg shorter leg p 3 y 3 3 units. Answer: The length y of the longer leg is Checkpoint 2. Find the unknown lengths x and y. Write your answers in simplest form. 15 60 x 30 y Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 219 30-60-90 Triangle In a 30-60-90 triangle, the shorter leg is opposite the 30 angle, and the longer leg is opposite the 60 angle. Words In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is the product of the length of the shorter . leg and 3 30 2a a 3 60 a Algebra hypotenuse 2 p shorter leg 2a longer leg shorter leg p 3 a3 Using a 30-60-90 Triangle Example 2 Find the length x of the hypotenuse and the length y of the longer leg of the triangle. 12 x 60 30 y The triangle is a 30-60-90 triangle. The length of the shorter leg is 12 units. a. hypotenuse 2 p shorter leg x 2 p 12 24 Answer: The length x of the hypotenuse is 24 units. b. longer leg shorter leg p 3 y 12 3 Answer: The length y of the longer leg is 12 3 units. Checkpoint 2. Find the unknown lengths x and y. Write your answers in simplest form. 15 60 x 30 y x 30, y 153 Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 219 Example 3 Using a Special Right Triangle An escalator going up to the second floor in a mall is 224 feet long and makes a 30 angle with the first floor. Find, to the nearest foot, the lengths of the triangle’s legs. 60 224 ft x 30 y Solution You need to find the length of the shorter leg first. 1. Find the length x of the shorter leg. hypotenuse 2 p shorter leg Rule for 30-60-90 triangle 2x Substitute. x Divide each side by . 2. Find the length y of the longer leg. longer leg shorter leg p 3 y 3 y≈ Rule for 30-60-90 triangle Substitute. Use a calculator. Answer: The length of the shorter leg is length of the longer leg is about 220 Chapter 9 • Pre-Algebra Notetaking Guide feet and the feet. Copyright © Holt McDougal. All rights reserved. Example 3 Using a Special Right Triangle An escalator going up to the second floor in a mall is 224 feet long and makes a 30 angle with the first floor. Find, to the nearest foot, the lengths of the triangle’s legs. 60 224 ft x 30 y Solution You need to find the length of the shorter leg first. 1. Find the length x of the shorter leg. hypotenuse 2 p shorter leg Rule for 30-60-90 triangle 224 2x Substitute. 112 x Divide each side by 2 . 2. Find the length y of the longer leg. longer leg shorter leg p 3 Rule for 30-60-90 triangle y 112 3 Substitute. y ≈ 194 Use a calculator. Answer: The length of the shorter leg is 112 feet and the length of the longer leg is about 194 feet. 220 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9.7 The Tangent Ratio Goal: Use tangent to find side lengths of right triangles. Vocabulary Trigonometric ratio: The Tangent Ratio The tangent of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. side opposite aA side adjacent to aA B hypotenuse c A a b tan A Example 1 side a opposite A C b side adjacent to A Finding a Tangent Ratio For T PQR, find the tangent of aP. opposite tan P adjacent P 39 R 89 80 Checkpoint 1. For T PQR in Example 1, find the tangent of aQ. Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 221 9.7 The Tangent Ratio Goal: Use tangent to find side lengths of right triangles. Vocabulary Trigonometric A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. ratio: The Tangent Ratio The tangent of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. side opposite aA side adjacent to aA B hypotenuse c A a b tan A Example 1 side a opposite A C b side adjacent to A Finding a Tangent Ratio For T PQR, find the tangent of aP. P 39 opposite 80 tan P 3 9 adjacent R 89 80 Checkpoint 1. For T PQR in Example 1, find the tangent of aQ. 39 80 Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 221 Using a Calculator Example 2 a. tan 24 Keystrokes When using a calculator to find a trigonometric ratio, make sure the calculator is in degree mode. Round the result to four decimal places if necessary. 2nd Display Answer Display Answer [TRIG] 24 ) b. tan 55 Keystrokes 2nd [TRIG] 55 ) Checkpoint Approximate the tangent value to four decimal places. 2. tan 5 3. tan 38 4. tan 72 Using a Tangent Ratio Example 3 Find the height h (in feet) of the roof to the nearest foot. h Solution 27 30 ft 30 ft Use the tangent ratio. In the diagram, the length of the leg opposite the 27 angle is h. The length of the adjacent leg is 30 feet. opposite adjacent tan 27 Definition of tangent ratio tan 27 Substitute. ≈ Use a calculator to approximate tan 27. ≈h Multiply each side by Answer: The height of the roof is about 222 Chapter 9 • Pre-Algebra Notetaking Guide . feet. Copyright © Holt McDougal. All rights reserved. Using a Calculator Example 2 a. tan 24 Keystrokes When using a calculator to find a trigonometric ratio, make sure the calculator is in degree mode. Round the result to four decimal places if necessary. 2nd Display [TRIG] 24 Answer 0.445228685 0.4452 ) b. tan 55 Keystrokes 2nd Display [TRIG] 55 Answer 1.428148007 1.4281 ) Checkpoint Approximate the tangent value to four decimal places. 2. tan 5 3. tan 38 0.0875 Example 3 4. tan 72 0.7813 3.0777 Using a Tangent Ratio Find the height h (in feet) of the roof to the nearest foot. h Solution 27 30 ft 30 ft Use the tangent ratio. In the diagram, the length of the leg opposite the 27 angle is h. The length of the adjacent leg is 30 feet. opposite adjacent tan 27 Definition of tangent ratio h tan 27 30 Substitute. h 0.5095 ≈ 30 15 ≈ h Use a calculator to approximate tan 27. Multiply each side by 30 . Answer: The height of the roof is about 15 feet. 222 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9.8 The Sine and Cosine Ratios Goal: Use sine and cosine to find triangle side lengths. The Sine and Cosine Ratios The sine of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. side opposite aA hypotenuse a c sin A B hypotenuse c A side a opposite A C b side adjacent to A The cosine of an acute angle of a right triangle is the ratio of the length of the angle’s adjacent side to the length of the hypotenuse. side adjacent aA hypotenuse b c cos A Finding Sine and Cosine Ratios Example 1 For T PQR, find the sine and cosine of aP. opposite sin P hypotenuse P 39 R 89 80 adjacent hypotenuse cos P Checkpoint 1. For T PQR in Example 1, find the sine and cosine of aQ. Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 223 9.8 The Sine and Cosine Ratios Goal: Use sine and cosine to find triangle side lengths. The Sine and Cosine Ratios The sine of an acute angle of a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. side opposite aA hypotenuse a c sin A B hypotenuse c A side a opposite A C b side adjacent to A The cosine of an acute angle of a right triangle is the ratio of the length of the angle’s adjacent side to the length of the hypotenuse. side adjacent aA hypotenuse b c cos A Finding Sine and Cosine Ratios Example 1 For T PQR, find the sine and cosine of aP. P 39 opposite 80 sin P 8 9 hypotenuse R 89 80 adjacent 39 cos P 8 9 hypotenuse Checkpoint 1. For T PQR in Example 1, find the sine and cosine of aQ. 39 89 80 89 sin Q , cos Q Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 223 Using a Calculator Example 2 a. sin 60 Keystrokes 2nd 60 Answer Display Answer [TRIG] ) Display b. cos 45 Keystrokes 2nd [TRIG] 45 ) Checkpoint Approximate the sine or cosine value to four decimal places. 2. cos 9 3. cos 78 4. sin 13 5. sin 88 Using a Cosine Ratio Example 3 Find the value of x in the triangle. In T DEF, DE &* is adjacent to aD. Because you know the length of the hypotenuse, use cos D and the definition of the cosine ratio to find the value of x. Round your answer to the nearest tenth of a unit. adjacent hypotenuse cos D 224 Chapter 9 • Pre-Algebra Notetaking Guide F 25 39 E D x Definition of cosine ratio Substitute. ≈ Use a calculator to approximate cos 39. ≈x Multiply each side by . Copyright © Holt McDougal. All rights reserved. Using a Calculator Example 2 a. sin 60 Keystrokes 2nd 60 [TRIG] ) Display Answer 0.866025404 0.8660 Display Answer 0.707106781 0.7071 b. cos 45 Keystrokes 2nd [TRIG] 45 ) Checkpoint Approximate the sine or cosine value to four decimal places. 2. cos 9 3. cos 78 4. sin 13 5. sin 88 0.2079 0.2250 0.9994 0.9877 Example 3 Using a Cosine Ratio Find the value of x in the triangle. In T DEF, DE &* is adjacent to aD. Because you know the length of the hypotenuse, use cos D and the definition of the cosine ratio to find the value of x. Round your answer to the nearest tenth of a unit. adjacent hypotenuse cos D x cos 39 25 x 25 0.7771 ≈ 19.4 ≈ x 224 Chapter 9 • Pre-Algebra Notetaking Guide F 25 39 E x D Definition of cosine ratio Substitute. Use a calculator to approximate cos 39. Multiply each side by 25 . Copyright © Holt McDougal. All rights reserved. Using a Sine Ratio Example 4 A ski jump is 140 meters long and makes an angle of 25 with the ground. To the nearest meter, estimate the height h of the ski jump. 140 m h 25 Solution To estimate the height of the ski jump, find the length of the side the 25 angle. Because you know the length of the hypotenuse, use sin 25. opposite hypotenuse sin 25 Definition of sine ratio sin 25 Substitute. ≈ Use a calculator to approximate sin 25. ≈h Multiply each side by Answer: The height of the ski jump is about Copyright © Holt McDougal. All rights reserved. . meters. Chapter 9 • Pre-Algebra Notetaking Guide 225 Example 4 Using a Sine Ratio A ski jump is 140 meters long and makes an angle of 25 with the ground. To the nearest meter, estimate the height h of the ski jump. 140 m h 25 Solution To estimate the height of the ski jump, find the length of the side opposite the 25 angle. Because you know the length of the hypotenuse, use sin 25. opposite hypotenuse sin 25 Definition of sine ratio h sin 25 140 Substitute. h 140 0.4226 ≈ 59 ≈ h Use a calculator to approximate sin 25. Multiply each side by 140 . Answer: The height of the ski jump is about 59 meters. Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 225 9 Words to Review Give an example of the vocabulary word. 226 Square root Perfect square Radical expression Simplest form of a radical expression Hypotenuse Leg Pythagorean theorem Acute triangle Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 9 Words to Review Give an example of the vocabulary word. Perfect square Square root A square root of a number n is the number m such that m2 n. Radical expression A number that is the square of an integer. 4 is a perfect square because 4 22. Simplest form of a radical expression An expression that involves a radical sign, . A radical expression is in simplest form when (1) no factor of the expression under the radical sign is a perfect square other than 1, and (2) there are no fractions under the radical sign and no radical sign in the denominator of any fraction. Leg Hypotenuse &* is the In T DEF, DF hypotenuse. &* and EF &* are legs. In T DEF, DE F F D E E D Pythagorean theorem For any right triangle, the sum of the squares of the lengths a and b of the legs equals the square of the length c of the hypotenuse: a2 b2 c 2. Acute triangle 53° 68° 59° 226 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Right triangle Obtuse triangle Equiangular triangle Irrational number Real number Distance formula Midpoint formula Slope formula Sine Cosine Copyright © Holt McDougal. All rights reserved. Chapter 9 • Pre-Algebra Notetaking Guide 227 Right triangle Obtuse triangle 58° 35° 124° 32° Equiangular triangle Irrational number A number that cannot be written as a quotient of two integers. The decimal form of an irrational number neither terminates nor repeats. 60° 60° 60° Real number Distance formula The set of all rational numbers and irrational numbers. For 1 example, 3, 0, , π, 5.17, 3 and 44 are all real numbers. Midpoint formula x1 x2 y1 y2 M , 2 2 Sine If points A(x1, y1) and B(x2, y2) do not lie on a vertical line, &( is then the slope of ^AB y2 y1 . x2 x1 P 5 3 R 4 opposite hypotenuse 3 5 sin Q Copyright © Holt McDougal. All rights reserved. (x2 x1)2 (y2 y1)2. d Cosine P R The distance d between two points (x1, y1) and (x2, y2) is given by Slope formula The coordinates of the midpoint M of a segment with endpoints A(x1, y1) and B(x2, y2) are given by 3 21° 5 4 adjacent hypotenuse 4 5 cos Q Chapter 9 • Pre-Algebra Notetaking Guide 227 Tangent Review your notes and Chapter 9 by using the Chapter Review on pages 526–529 of your textbook. 228 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Tangent P 3 R 5 4 opposite adjacent 3 4 tan Q Review your notes and Chapter 9 by using the Chapter Review on pages 526–529 of your textbook. 228 Chapter 9 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved.