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Real Numbers and Right Triangles CHAPTER Real Numbers and Right Triangles BEFORE In previous chapters you’ve . • Used ratios and proportions • Solved problems using similar triangles • Found the slope of a line through two points Now In Chapter 9 you’ll study . • Using square roots • Solving problems using the Pythagorean theorem • Comparing and ordering real numbers • Using the distance, midpoint, and slope formulas • Applying the tangent, sine, and How do architects cosine ratios use right triangles? WHY? So you can solve real-world problems about . • photography, p. 480 • walking speed, p. 484 • synchronized swimming, p. 492 • maps, p. 506 • stereo speakers, p. 512 • forestry, p. 518 474 PAFL_229843_C09_CO.indd 474 12/24/08 3:14:16 PM Chapter 9 Resources Go to thinkcentral.com • Online Edition • eWorkbook • @HomeTutor • State Test Practice • More Examples • Video Tutor MATH In the Real World Courtyard Each of the triangles in this courtyard in Los Angeles, California, is a right triangle. In this chapter, you will use special relationships among the side lengths of right triangles to solve problems about architecture. What do you think? To create triangles like those in the photo, draw a square on graph paper. Then draw a diagonal of the square. Do the two triangles appear to be congruent? Explain. PAFL_229843_C09_CO.indd 475 12/24/08 3:14:23 PM CHAPTER 3 Chapter Prerequisite Skills PREREQUISITE SKILLS QUIZ Review Vocabulary Preparing for Success To prepare for success in this chapter, test your rational number, p. 221 knowledge of these concepts and skills. You may want to look at the terminating decimal, p. 221 pages referred to in blue for additional review. repeating decimal, p. 221 ratio, p. 275 1. Vocabulary Is the number Ϫ3 a rational number? an integer? a whole similar figures, p. 300 number? slope, p. 420 17 16 2. Order the numbers Ϫᎏᎏ, Ϫ2, ᎏᎏ, and Ϫ3.7 from least to greatest. 9 3 (p. 221) Tell whether the ratio is in simplest form. If it is not, write it in simplest form. (p. 275) 34 23 3. 3 to 6 4. ᎏ 5. ᎏ 6. 76:38 51 3 7. Shadows A student who is 5 feet tall is standing next to a fence post. The student casts a shadow 15 feet long. The fence post casts a shadow 18 feet long. How tall is the fence post? (p. 305) Find the slope of the line through the given points. (p. 420) 8. (Ϫ2, Ϫ8), (7, 3) 9. (4, 5), (4, 2) 10. (Ϫ5, 3), (Ϫ10, 13) NOTETAKING STRATEGIES USING A CONCEPT MAP When you learn new concepts that Note Worthy are related to each other, you may find it helpful to organize the ideas in a concept map. You will find a notetaking strategy at the beginning Rational numbers of each chapter. Look for additional notetaking and study strategies throughout the chapter. Fractions Decimals Integers Terminating Repeating Whole numbers In Lesson 9.6, you can use a concept map to help you organize your notes on triangles. 476 PAFL_229843_C09_SG.indd 476 12/24/08 3:14:31 PM 9 1 Square Roots LESSON . BEFORE Now WHY? You found squares of You’ll find and approximate So you can find a person’s Vocabulary numbers. square roots of numbers. running speed, as in Ex. 59. square root, p. 477 perfect square, p. 478 radical expression, p. 478 Human Chess In September of every even-numbered year, people in Marostica, Italy, play an unusual chess game. Each chess piece is portrayed by a person. The people portraying the knights are even on horseback! The chessboard is a square with an area of 324 square meters. What is the length of each side of the board? To answer this question, you need to find the square root of 324. A square root of a number n is a number m such that m2 ϭ n. Every positive number has two square roots. One square root is positive ᎏ and the other is negative. The radical sign, ͙a , represents a nonnegative square root. The symbol Ϯ, read “plus or minus,” refers to both square roots of a positive number. For example: ᎏ ͙100 ϭ 10 Positive square root of 100 ᎏ Ϫ ͙100 ϭ Ϫ10 Negative square root of 100 ᎏ Ϯ ͙100 ϭ Ϯ10 Positive or negative square root of 100 Zero has only one square root, itself. Example 1 Finding a Square Root Study Strategy The chessboard described above is a square with an area of 324 square meters, so the length of each side of the chessboard is the In Example 1, it doesn’t make positive square root of 324. sense to find the negative ᎏ ͙ ϭ 2 square root of 324, because 324 18 because 18 324. length cannot be negative. Answer The length of each side of the chessboard is 18 meters. Checkpoint Find the square roots of the number. 1. 16 2. 64 3. 144 4. 256 Video Tutor Go to thinkcentral.com Lesson 9.1 Square Roots 477 PANL_9780547587776_C09L01.indd 477 11/16/10 9:54:17 AM Approximating Square Roots A perfect square is a number that is the square of an integer. For example, 1, 4, and 9 are perfect squares. Note Worthy 1 ϭ 12, 4 ϭ 22, and 9 ϭ 32 You can use perfect squares to approximate a square root of a number. You may find it helpful to make a list of the first 20 perfect squares and their square roots in your notebook and Example 2 Approximating a Square Root memorize them. You can find ᎏ a table of squares and square Approximate ͙51 to the nearest integer. roots on p. 798. The perfect square closest to, but less than, 51 is 49. The perfect square closest to, but greater than, 51 is 64. So, 51 is between 49 and 64. This statement can be expressed by the compound inequality 49 < 51 < 64. 49 < 51 < 64 Identify perfect squares closest to 51. ᎏ ᎏ ᎏ ͙49 < ͙51 < ͙64 Take positive square root of each number. ᎏ 7 < ͙51 < 8 Evaluate square root of each perfect square. 2 ϭ Answer The ᎏaverage of 7 and 8 is 7.5, and (7.5) 56.25. Becauseᎏ 51 < 56.25, ͙51 is closer to 7 than to 8. So, to the nearest integer, ͙51 ഠ 7. Checkpoint ᎏ 5. Approximate ͙125 to the nearest whole number. Example 3 Using a Calculator ᎏ Tech Help Use a calculator to approximate ͙ 515 . Round to the nearest tenth. When you enter a square root Keystrokes Answer on the calculator, you should ᎏ ᎏ ͙ ͙ ഠ also enter a right parenthesis [ a ] 515 515 22.7 to close the left parenthesis that the calculator enters. Although the calculator shows 8 digits for the decimal part of Radical Expressions A radical expression is an expression that involves the square root in Example 3, the decimal actually continues a radical sign. The horizontal bar in a radical sign is a grouping symbol. without end. When you evaluate a radical expression, evaluate the expression inside the radical symbol before finding the square root. Example 4 Evaluating a Radical Expression ᎏ .5 ؍ and b 11 ؍ Evaluate 2 ͙ a ؉ b 2 when a ᎏ ᎏ 2 ͙ a ϩ b 2 ϭ 2 ͙ 11 5 2 Substitute 11 for a and 5 for b. ᎏ ϭ 2 ͙36 Evaluate expression inside radical symbol. ϭ 2 p 6 ϭ 12 Evaluate square root. Multiply. 478 Chapter 9 Real Numbers and Right Triangles PANL_9780547587776_C09L01.indd 478 11/6/10 10:39:29 AM Checkpoint .4 ؍ and b 12 ؍ Evaluate the expression when a ᎏ ᎏ ᎏ 6. ͙a ϩ b 7. ͙ b 2 Ϫ a 8. 3 ͙ab ϩ 1 Example 5 Solving an Equation Using Square Roots Physics An amusement park ride includes a free fall drop of 272 feet. You can use the equation d ϭ 16t 2 to determine the time t in seconds that it takes a dropped object to fall a distance of d feet. How long does the free fall part of the ride take? Solution d ϭ 16t 2 Write original equation. 272 ϭ 16t 2 Substitute 272 for d. 17 ϭ t 2 Divide each side by 16. ᎏ Ϯ ͙17 ϭ t Use definition of square root. Ϯ4.1 ഠ t Use a calculator to approximate square root. Answer Because only the positive solution makes sense in this situation, the free fall part of the ride takes about 4.1 seconds. 9.1 Exercises More Practice, p. 787 Go to thinkcentral.com Practice Exercises Guided Practice Vocabulary Check 1. Describe and give an example of a perfect square. 2. You know that one square root of a number x is Ϫ9. What is the other square root? What is the value of x? Skill Check Find the square roots of the number. 3. 4 4. 36 5. 121 6. 225 Approximate the square root to the nearest integer. ᎏ ᎏ ᎏ ᎏ 7. ͙10 8. Ϫ ͙84 9. ͙151 10. Ϫ ͙200 Solve the equation. 11. a2 ϭ 9 12. n2 ϭ 25 13. 361 ϭ x 2 14. 400 ϭ y 2 Guided Problem Solving 15. Eiffel Tower The base of the Eiffel Tower is a square with an area of 15,625 square feet. What is the length of a side of the base? 1 Write an equation that relates base area A and side length s. 2 Substitute 15,625 for A in the equation in Step 1 and solve for s. Lesson 9.1 Square Roots 479 PANL_9780547587776_C09L01.indd 479 11/6/10 10:39:33 AM Practice and Problem Solving In the following exercises, you may find it helpful to use a calculator for Homework Help approximating square roots.
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