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8-1 Exercises 8-1 Exercises KEYWORD: MG7 8-1 KEYWORD: MG7 Parent GUIDED PRACTICE Assignment Guide 8 1. Vocabulary In the proportion __28 = __ , which number is the geometric mean of 32 the other two numbers? 8 is the geometric mean of 2 and 32. SEE EXAMPLE 1 p. 518 Write a similarity statement comparing the three triangles in each diagram. 2. 3. * XYZ ∼ XWY ∼ YWZ 4. 8 7 , SEE EXAMPLE SEE EXAMPLE + - BED ∼ ECD ∼ BCE RPQ ∼ PSQ ∼ RSP < 9 2 Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1 and 8 2 5. 2 and 50 10 6. 4 and 16 8 7. _ 2 3 77 8. 9 and 12 6 √ 9. 16 and 25 20 10. 7 and 11 √ 3 Find x, y, and z. p. 519 p. 520 11. È p. 520 12. { Þ Ý SEE EXAMPLE 4 Assign Guided Practice exercises as necessary. â 15 ; 2 √ 2 √6; 2 √10 â Þ £ä 5; 10 √5; 13. 5 √ 5 Þ If you finished Examples 1–4 Basic 15–37, 41, 44, 47–49, 54–62 Average 15–27, 30–38 even, 39–50, 54–62 Advanced 15–27, 28–40 even, 42–62 ÈÊÊȖв £ÎÊ вÊ Ý Homework Quick Check Ý â Óä £n 13 ; 8 12; 4 √ 14. Measurement To estimate the length of the USS Constitution in Boston harbor, a student locates points T and U as shown. What is RS to the nearest tenth? 62.0 m , 1 / {Ê ÈäÊ PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 15–17 18–23 24–26 27 1 2 3 4 Extra Practice Skills Practice p. S18 Application Practice p. S35 Write a similarity statement comparing the three triangles in each diagram. 15. 16. + RSU ∼ RTS ∼ STU 17. , / * MPN ∼ PQN ∼ MQP If you finished Examples 1–2 Basic 15–23, 28, 29 Average 15–23, 28, 29, 45 Advanced 15–23, 28, 29, 45, 46 CAB ∼ ADB ∼ CDA - 1 Quickly check key concepts. Exercises: 16, 18, 24, 27, 36 Visual In Exercises 2–4 and 15–17, some students may have difficulty visualizing the corresponding angles in the overlapping triangles. Suggest that they redraw the diagram to separate the three triangles. Inclusion Remind students not to rely on the given diagram for scale in Exercises 14 and 27. Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 18. 5 and 45 15 1 and 80 2 √ 5 21. _ 4 10 20. 5 and 8 2 √ 27 2 _ _ 3 √ 5 and 23. 3 40 5 19. 3 and 15 3 √ 2 22. 1.5 and 12 3 √ _ 10 Find x, y, and z. 24. 25. { £Ó Þ Ý Ý £Ó°n â Þ â 32; 8 √2; 24 √ 2 12; 16; 7.2 26. Þ °È Îä â {ä 20 √ 3 ; 10 √ 21 ; 20 √ 7 Ý 8-1 Similarity in Right Triangles ge07se_c08_0518_0523.indd 521 521 9/19/05 4:38:27 PM KEYWORD: MG7 Resources Lesson 8-1 521 Algebra Before students begin Exercise 29, you may want to review the properties of exponents. 2 2 That is, (a √x) = a 2( √ x ) = a 2x, for x > 0. 27. Measurement To estimate the height of the Taipei 101 tower, Andrew stands so that his lines of sight to the top and bottom of the tower form a 90° angle. What is the height of the tower to the nearest foot? 1670 ft 91 ft 3 in. 28. The geometric mean of two numbers is 8. One of the numbers is 2. Find the other number. 32 _ 5. 29. The geometric mean of two numbers is 2 √ 10 , One of the numbers is 6. Find the other number. 3 Number Sense If the average rate of return in Exercise 40 were the arithmetic mean, it would be 6.5%. Unfortunately for the consumer, the geometric mean of two numbers is always smaller than the arithmetic mean. 5 ft _ or 3 1 3 Use the diagram to complete each equation. x+y _ ? u v x =_ z y _ 31. _ 32. _ 30. _ z u=xx+y v = ? y ? y 2 z 33. _ = _ 34. (?) = y(x + y) v 35. u 2 = (x + y)(?) x x z ? Ý Õ Þ â Û Give each answer in simplest radical form. 36. AD = 12, and CD = 8. Find BD. 4 √ 6 Exercise 44 involves using the geometric mean corollaries to find the lengths of a guy wire anchor. This exercise prepares students for the Multi-Step Test Prep on page 542. 37. AC = 16, and CD = 5. Find BC. 4 √ 5 38. AD = CD = √ 2 . Find BD. , and AC = √ 39. BC = √5 10 . Find CD. 41. _ _ 45. The area of the rect. is ab, and the area of the square is s 2. It is given that s 2 = ab, so s is the geometric mean of a and b. 46. Let z be the geometric mean of x and y, where x = a 2 and y = b 2. 2 2 b = ab, which is a So z = √a whole number. 52. Let AD = DC = a. By Corollary 8-1-3, AB 2 = (a)(2a) = 2a 2, and BC 2 = (a)(2a) = 2a 2. So AB = 2 . Therefore ABC is BC = a √ isosceles, so it is a 45°-45°-90° triangle. √ 10 _ 2 40. Finance An investment returns 3% one year and 10% the next year. The average rate of return is the geometric mean of the two annual rates. What is the average rate of return for this investment to the nearest tenth of a percent? 5.5% 41. B; the proportion should be 12 = EF . EF 8 Answers √ 2 /////ERROR ANALYSIS///// Two students were asked to find EF. Which solution is incorrect? Explain the error. ! 43. By Corollary 8-1-3, a 2 = x(x + y), and b 2 = y(x + y). So a 2 + b 2 = x(x + y) + y(x + y). 42. By the Distrib. Prop., this expression simplifies to (x + y) (x + y) = (x + y)2 43. = c 2. So a 2 + b 2 = c 2. " £Ó ÊÚÚÚ ÊÊÊÊÊÊÚÚÚ ÊÊÊÊ]ÊÃÊÊ ÚÚÚ ÊnÊÊÊÊÊÊÚÚÚ ÊÊÊÊ]ÊÃÊÊ n Ó Ê®Ê ÊÊ£Ó®n®ÊÊÈ°ÊÊ Ê е / ÕÃÊÊÊÊÊ ȖÊ ÈÊ еÊ Ê Ê еее еÊ Ê°Ê Ê ÊÊ{Ê ÊȖÈÊ ȖÊÊÊ£È®È®Ê n { Ó Ê®Ê ÊÊn®{®ÊÊÎÓ°ÊÊ Ê / ÕÃÊÊÊÊÊ ȖÊ е ÎÓÊ еÊ Ê Ê еее еÊ Ê° Ê ÊÊ{Ê ÊȖÓÊ ȖÊÊÊ£È®Ó®Ê { The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments that are 2 cm long and 5 cm long. Find the length of the altitude to the nearest tenth of a centimeter. 3.2 cm Critical Thinking Use the figure to show how Corollary 8-1-3 can be used to derive the Pythagorean Theorem. (Hint: Use the corollary to write expressions for a 2 and b 2. Then add the expressions.) Ý V > Þ L 44. This problem will prepare you for the Multi-Step Test Prep on page 542. Before installing a utility pole, a crew must first dig a hole and install the anchor for the guy wire −−− −− that supports the pole. In the diagram, SW ⊥ RT, −−− −−− RW ⊥ WT, RS = 4 ft, and ST = 3 ft. −−− a. Find the depth of the anchor SW to the nearest inch. 3 ft 6 in. −−− b. Find the length of the rod RW to the nearest inch. 8-1 PRACTICE A 8-1 PRACTICE B n£ 3IMILARITYIN2IGHT4RIANGLES 522 7RITEASIMILARITYSTATEMENTCOMPARINGTHETHREETRIANGLESINEACHDIAGRAM - * + $ 8 ,%33/. % , & ' 7 9 O$%&O'%$ O789O:87 O,+- O'$& O:79 AND ?? ????? q AND ^ q X 5 10 Z Z ^ ^ qqq 15 Y 6 X ^ ge07se_c08_0518_0523.indd $IAGRAMS 522 ^ Y X 27 %QUATIONS 4HELENGTHOFTHE ALTITUDEISTHEGEOMETRIC MEANOFTHELENGTHSOF THETWOSEGMENTSOFTHE HYPOTENUSE Y ?? ?? 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E B !LTITUDEAND (YPOTENUSE qi 6 Y 7 ^ 8-1 READING STRATEGIES 5SEA4ABLE 7ORDS X ,i>`}Ê-ÌÀ>Ìi}ià X ^ q &INDXYANDZ Y n£ 5SETHETABLEBELOWTOUNDERSTANDTHERELATIONSHIPSFORMEDWHENTHEALTITUDE ISDRAWNFROMTHEVERTEXOFARIGHTTRIANGLETOITSHYPOTENUSE " &INDTHEGEOMETRICMEANOFEACHPAIROFNUMBERS)FNECESSARYGIVETHE ANSWERINSIMPLESTRADICALFORM AND ?? AND AND ^ 7 Chapter 8 Right Triangles and Trigonometry : 0OSSIBLEANSWERS O*+,O*,- / 5 ft 3 in. 8-1 PRACTICE C - *À>VÌViÊ ,%33/. , C Y AND AND ^ q ^ q