6 Similarity. Prerequisite Skills. Before VOCABULARY CHECK SKILLS AND ALGEBRA CHECK

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1 6 Similarit 6. atios, roportions, and the Geometric Mean 6.2 Use roportions to Solve Geometr roblems 6.3 Use Similar olgons 6.4 rove Triangles Similar b 6.5 rove Triangles Similar b SSS and SS 6.6 Use roportionalit Theorems 6.7 erform Similarit Transformations efore In previous courses and in hapters 5, ou learned the following skills, which ou ll use in hapter 6: using properties of parallel lines, using properties of triangles, simplifing epressions, and finding perimeter. rerequisite Skills VOULY HK. The alternate interior angles formed when a transversal intersects two? lines are congruent. 2. Two triangles are congruent if and onl if their corresponding parts are?. SKILLS N LG HK Simplif the epression. (eview pp. 870, 874 for 6..) 3. 9 p 20 } } } Ï } 5(5 p 7) ind the perimeter of the rectangle with the given dimensions. (eview p. 49 for 6., 6.2.) 7. l 5 5 in., w 5 2 in. 8. l 5 30 ft, w 5 0 ft m 2, l 5 8 m 0. ind the slope of a line parallel to the line whose equation is ( 2). (eview p. 7 for 6.5.) 354

2 Now In hapter 6, ou will appl the big ideas listed below and reviewed in the hapter Summar on page 47. You will also use the ke vocabular listed below. ig Ideas Using ratios and proportions to solve geometr problems 2 Showing that triangles are similar 3 Using indirect measurement and similarit KY VOULY ratio, p. 356 proportion, p. 358 means, etremes geometric mean, p. 359 scale drawing, p. 365 scale, p. 365 similar polgons, p. 372 scale factor of two similar polgons, p. 373 dilation, p. 409 center of dilation, p. 409 scale factor of a dilation, p. 409 reduction, p. 409 enlargement, p. 409 Wh? You can use similarit to measure lengths indirectl. or eample, ou can use similar triangles to find the height of a tree. Geometr The animation illustrated below for ercise 33 on page 394 helps ou answer this question: What is the height of the tree? You can use proportional reasoning to estimate the height of a tall tree. Use similar triangles to write a proportion. Then find the value of. Geometr at classzone.com Geometr at classzone.com Other animations for hapter 6: pages 365, 375, 39, 407, and 44 Other animations for hapter appear on pages 7, 9, 4, 2, 37, and

3 6. atios, roportions, and the Geometric Mean efore You solved problems b writing and solving equations. Now You will solve problems b writing and solving proportions. Wh? So ou can estimate bird populations, as in. 62. Ke Vocabular ratio proportion means, etremes geometric mean If a and b are two numbers or quantities and b Þ 0, then the ratio of a to b is a } b. The ratio of a to b can also be written as a : b. or eample, the ratio of a side length in n to a side length in n can be written as } 2 or 2:. 2 m 2 m 2 m m m m atios are usuall epressed in simplest form. Two ratios that have the same simplified form are called equivalent ratios. The ratios 7 : 4 and : 2 in the eample below are equivalent. U 4 ft S 7 ft T width of STU }} 5 } 7 ft 5 } length of STU 4 ft 2 XML Simplif ratios Simplif the ratio. a. 64 m:6 m b. 5 ft } 20 in. VIW UNIT NLYSIS or help with measures and conversion factors, see p. 886 and the Table of Measures on p. 92. Solution a. Write 64 m : 6 m as 64 m } 6 m. Then divide out the units and simplif. 64 m } 5 } :3 6 m 3 b. To simplif a ratio with unlike units, multipl b a conversion factor. 5 ft } 5 5 ft 2 in. } p} 5 } 60 5 } 3 20 in. 20 in. ft 20 GUI TI for ample Simplif the ratio.. 24 ards to 3 ards cm : 6 m 356 hapter 6 Similarit

4 XML 2 Use a ratio to find a dimension INTING You are planning to paint a mural on a rectangular wall. You know that the perimeter of the wall is 484 feet and that the ratio of its length to its width is 9 : 2. ind the area of the wall. WIT XSSIONS ecause the ratio in ample 2 is 9 : 2, ou can write an equivalent ratio to find epressions for the length and width. length } 5 } 9 width } 2 p } 5 9 } 2. Solution ST Write epressions for the length and width. ecause the ratio of length to width is 9 : 2, ou can represent the length b 9 and the width b 2. ST 2 Solve an equation to find. 2l 2w 5 ormula for perimeter of rectangle 2(9) 2(2) Substitute for l, w, and Multipl and combine like terms ivide each side b 22. ST 3 valuate the epressions for the length and width. Substitute the value of into each epression. Length (22) 5 98 Width (22) 5 44 c The wall is 98 feet long and 44 feet wide, so its area is 98 ft p 44 ft ft 2. XML 3 Use etended ratios LG The measures of the angles in n are in the etended ratio of :2:3. ind the measures of the angles. Solution 8 egin b sketching the triangle. Then use the etended ratio of :2:3 to label the measures as8, 28, and Triangle Sum Theorem ombine like terms ivide each side b 6. c The angle measures are 308, 2(308) 5 608, and 3(308) GUI TI for amples 2 and 3 3. The perimeter of a room is 48 feet and the ratio of its length to its width is 7 : 5. ind the length and width of the room. 4. triangle s angle measures are in the etended ratio of :3:5. ind the measures of the angles. 6. atios, roportions, and the Geometric Mean 357

5 OOTIONS n equation that states that two ratios are equal is called a proportion. etreme mean a } b 5 c } d mean etreme The numbers b and c are the means of the proportion. The numbers a and d are the etremes of the proportion. The propert below can be used to solve proportions. To solve a proportion, ou find the value of an variable in the proportion. KY ONT or Your Notebook OOTIONS You will learn more properties of proportions on p ropert of roportions. ross roducts ropert In a proportion, the product of the etremes equals the product of the means. If a } b 5 c } d where b Þ 0andd Þ 0, then ad 5 bc. 2 } p } p XML 4 Solve proportions a. LG Solve the proportion. 5 } 0 5 } 6 b. } 5 2 } 3 Solution NOTH WY In part (a), ou could multipl each side b the denominator, 6. Then 6 p } p }, 0 6 so 8 5. a. b. 5 } 0 5 } 6 Write original proportion. 5 p p ross roducts ropert Multipl. 8 5 ivide each side b 0. } 5 2 } 3 Write original proportion. p 3 5 2( ) ross roducts ropert istributive ropert Subtract 2 from each side. GUI TI for ample 4 Solve the proportion } 5 5 } 8 6. } } } 7 5 } hapter 6 Similarit

6 XML 5 Solve a real-world problem SIN s part of an environmental stud, ou need to estimate the number of trees in a 50 acre area. You count 270 trees in a 2 acre area and ou notice that the trees seem to be evenl distributed. stimate the total number of trees. Solution Write and solve a proportion involving two ratios that compare the number of trees with the area of the land. 270 } 5 n number of trees } Write proportion area in acres 270 p p n ross roducts ropert 20,250 5 n Simplif. c There are about 20,250 trees in the 50 acre area. KY ONT or Your Notebook Geometric Mean The geometric mean of two positive numbers a and b is the positive number that satisfies } a 5 }. So, 2 5 ab and 5 Ï } ab. b XML 6 ind a geometric mean ind the geometric mean of 24 and 48. Solution 5 Ï } ab efinition of geometric mean 5 Ï } 24 p 48 Substitute 24 for a and 48 for b. 5 Ï } 24 p 24 p 2 actor. 5 24Ï } 2 Simplif. c The geometric mean of 24 and 48 is 24Ï } 2 ø GUI TI for amples 5 and 6 8. WHT I? In ample 5, suppose ou count 390 trees in a 3 acre area of the 50 acre area. Make a new estimate of the total number of trees. ind the geometric mean of the two numbers and and and 8 6. atios, roportions, and the Geometric Mean 359

7 6. XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s. 5, 27, and 59 5 STNIZ TST TI s. 2, 47, 48, 52, and 63 5 MULTIL SNTTIONS. 66. VOULY op the proportion } m 5 } p. Identif the means of the n q proportion and the etremes of the proportion. 2. WITING Write three ratios that are equivalent to the ratio 3 : 4. plain how ou found the ratios. XML on p. 356 for s. 3 7 SIMLIYING TIOS Simplif the ratio. 3. $20:$ cm 2 } 2 cm L : 0 ml 6. mi } 20 ft 7. 7 ft } 2 in cm } 2 m 9. 3 lb } 0 oz 0. 2 gallons } 8 quarts WITING TIOS ind the ratio of the width to the length of the rectangle. Then simplif the ratio.. 5 in. 5 in cm 8 cm 3. 0 m 320 cm INING TIOS Use the number line to find the ratio of the distances } 5. } 6. } 7. } XML 2 on p. 357 for s. 8 9 XML 3 on p. 357 for s IMT The perimeter of a rectangle is 54 feet. The ratio of the length to the width is 0 :. ind the length and the width. 9. SGMNT LNGTHS In the diagram, : is 2:7 and ind and. USING XTN TIOS The measures of the angles of a triangle are in the etended ratio given. ind the measures of the angles of the triangle :5:0 2. 2:7:9 22. :2:3 36 XML 4 on p. 358 for s LG Solve the proportion. 6 } 5 3 } } } } } z 26. j } } } c } } a } b } } } 2p 5 5 } 9p 360 hapter 6 Similarit

8 XML 6 on p. 359 for s GOMTI MN ind the geometric mean of the two numbers and and and and and and O NLYSIS student incorrectl simplified the ratio. escribe and correct the student s error. 8 in. } 5 } 8 in. p 2 in. 96 in. } 5} 5 3 ft 3 ft ft 3 ft 32 in. } ft WITING TIOS Let 5 0, 5 3, and z 5 8. Write the ratio in simplest form. 38. : z LG Solve the proportion. 2 5 } 5 } } s } 5 2s } } 2 2z 4. 5 } m 5 m } z } 3 7 } 5 q } 2 q NGL MSUS The ratio of the measures of two supplementar angles is 5 : 3. ind the measures of the angles. 47. SHOT SONS The ratio of the measure of an eterior angle of a triangle to the measure of the adjacent interior angle is : 4. Is the triangle acute or obtuse? plain how ou found our answer. 48. SHOT SONS Without knowing its side lengths, can ou determine the ratio of the perimeter of a square to the length of one of its sides? plain. LG In ercises 49 5, the ratio of two side lengths for the triangle is given. Solve for the variable. 49. : is 3: : is 3:4. 5. : is 5:9. p q 5 5 5r MULTIL HOI What is a value of that makes } } 3 true? The area of a rectangle is 4320 square inches. The ratio of the width to the length is 5 : 6. ind the length and the width. 54. OOINT GOMTY The points (23, 2), (, ), and (, 0) are collinear. Use slopes to write a proportion to find the value of. 55. LG Use the proportions a b } 2a 2 b 5 5 } 4 and b } a } 9 to find a and b. 56. HLLNG ind the ratio of to given that 5 } 7 } 5 24 and 2 } 2 } atios, roportions, and the Geometric Mean 36

9 OLM SOLVING XML 2 on p. 357 for TILING The perimeter of a room is 66 feet. The ratio of its length to its width is 6 : 5. You want to tile the floor with 2 inch square tiles. ind the length and width of the room, and the area of the floor. How man tiles will ou need? The tiles cost $.98 each. What is the total cost to tile the floor? 58. GS The gear ratio of two gears is the ratio of the number of teeth of the larger gear to the number of teeth of the smaller gear. In a set of three gears, the ratio of Gear to Gear is equal to the ratio of Gear to Gear. Gear has 36 teeth and Gear has 6 teeth. How man teeth does Gear have? 59. TIL MIX You need to make 36 one-half cup bags of trail mi for a class trip. The recipe calls for peanuts, chocolate chips, and raisins in the etended ratio 5::4. How man cups of each item do ou need? 60. SIZS International standard paper sizes are commonl used all over the world. The various sizes all have the same width-tolength ratios. Two sizes of paper are shown, called 3 and 2. The distance labeled is the geometric mean of 297 mm and 594 mm. ind the value of mm mm mm mm 6. TTING VG The batting average of a baseball plaer is the ratio of the number of hits to the number of official at-bats. In 2004, Johnn amon of the oston ed So had 62 official at-bats and a batting average of.304. Use the proportion to find the number of hits made b Johnn amon. Number of hits atting average }} 5}} Number of at-bats.000 XML 5 on p. 359 for MULTI-ST OLM The population of ed-tailed hawks is increasing in man areas of the United States. One long-term surve of bird populations suggests that the ed-tailed hawk population is increasing nationall b 2.7% each ear. a. Write 2.7% as a ratio of hawks in ear n to hawks in ear (n 2 ). b. In 2004, observers in orpus hristi, TX, spotted 80 migrating edtailed hawks. ssuming this population follows the national trend, about how man ed-tailed hawks can the epect to see in 2005? c. Observers in Lipan oint, Z, spotted 95 migrating ed-tailed hawks in ssuming this population follows the national trend, about how man ed-tailed hawks can the epect to see in 2006? WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI 5 MULTIL SNTTIONS

10 63. SHOT SONS Some common computer screen resolutions are 024:768, 800:600, and 640:480.plain wh these ratios are equivalent. 64. IOLOGY The larvae of the Mother-of-earl moth is the fastest moving caterpillar. It can run at a speed of 5 inches per second. When threatened, it can curl itself up and roll awa 40 times faster than it can run. How fast can it run in miles per hour? How fast can it roll? 65. UNY XHNG mil took 500 U.S. dollars to the bank to echange for anadian dollars. The echange rate on that da was.2 anadian dollars per U.S. dollar. How man anadian dollars did she get in echange for the 500 U.S. dollars? 66. MULTIL SNTTIONS Let and be two positive numbers whose geometric mean is 6. a. Making a Table Make a table of ordered pairs (, ) such that Ï } 5 6. b. rawing a Graph Use the ordered pairs to make a scatter plot. onnect the points with a smooth curve. c. nalzing ata Is the data linear? Wh or wh not? 67. LG Use algebra to verif ropert, the ross roducts ropert. 68. LG Show that the geometric mean of two numbers is equal to the arithmetic mean (or average) of the two numbers onl when the numbers are equal. (Hint: Solve Ï } 5 } with, 0.) 2 HLLNG In ercises 69 7, use the given information to find the value(s) of. ssume that the given quantities are nonnegative. 69. The geometric mean of the quantities (Ï } ) and (3Ï } ) is ( 2 6). 70. The geometric mean of the quantities ( ) and (2 3) is ( 3). 7. The geometric mean of the quantities (2 ) and (6 ) is (4 2 ). MIX VIW VIW repare for Lesson 6.2 in s ind the reciprocal. (p. 869) } } Solve the quadratic equation. (p. 882) ( 2 3)( 3) 5 27 Write the equation of the line with the given description. (p. 80) 79. arallel to , passing through (, 2) 80. erpendicular to 5 } 5, passing through (0, 24) 4 XT TI for Lesson 6., p ONLIN QUIZ at classzone.com 363

11 6.2 Use roportions to Solve Geometr roblems efore You wrote and solved proportions. Now You will use proportions to solve geometr problems. Wh? So ou can calculate building dimensions, as in. 22. Ke Vocabular scale drawing scale In Lesson 6., ou learned to use the ross roducts ropert to write equations that are equivalent to a given proportion. Three more was to do this are given b the properties below. KY ONT or Your Notebook VIW IOLS or help with reciprocals, see p dditional roperties of roportions 2. eciprocal ropert If two ratios are equal, then their reciprocals are also equal. 3. If ou interchange the means of a proportion, then ou form another true proportion. 4. In a proportion, if ou add the value of each ratio s denominator to its numerator, then ou form another true proportion. If a } b 5 c } d, then b } a 5 d } c. If a } b 5 c } d, then a } c 5 b } d. If a } b 5 c } d, then a b } b 5 c d } d. XML Use properties of proportions In the diagram, } MN 5 } N. S ST Write four true proportions. Solution ecause MN } S 5 N } ST, then 8 } }. M 8 N 4 0 S T the eciprocal ropert, the reciprocals are equal, so 0 } 8 5 } 4. ropert 3, ou can interchange the means, so 8 } }. ropert 4, ou can add the denominators to the numerators, so 8 0 } 5 4 }, or } } hapter 6 Similarit

12 XML 2 Use proportions with geometric figures LG In the diagram, } 5 }. ind and. Solution } 5 } Given } 5} ropert of roportions (ropert 4) } 5 8 } 6 Substitution ropert of qualit (8 6) ross roducts ropert 5 2 Solve for. c So, 5 2 and at classzone.com SL WING scale drawing is a drawing that is the same shape as the object it represents. The scale is a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object. XML 3 ind the scale of a drawing LUINTS The blueprint shows a scale drawing of a cell phone. The length of the antenna on the blueprint is 5 centimeters. The actual length of the antenna is 2 centimeters. What is the scale of the blueprint? Solution To find the scale, write the ratio of a length in the drawing to an actual length, then rewrite the ratio so that the denominator is. length on blueprint }} 5 } 5 cm 5 5 } } 2.5 length of antenna 2 cm c The scale of the blueprint is 2.5 cm : cm. GUI TI for amples, 2, and 3. In ample, find the value of. 2. In ample 2, } 5 }. ind. 3. WHT I? In ample 3, suppose the length of the antenna on the blueprint is 0 centimeters. ind the new scale of the blueprint. 6.2 Use roportions to Solve Geometr roblems 365

13 XML 4 Use a scale drawing MS The scale of the map at the right is inch : 26 miles. ind the actual distance from ocahontas to lgona. Solution Use a ruler. The distance from ocahontas to lgona on the map is about.25 inches. Let be the actual distance in miles..25 in. } 5} in. mi 26 mi 5.25(26) distance on map actual distance ross roducts ropert Simplif. c The actual distance from ocahontas to lgona is about 32.5 miles. XML 5 Solve a multi-step problem SL MOL You bu a 3- scale model of the eunion Tower in allas, TX. The actual building is 560 feet tall. Your model is 0 inches tall, and the diameter of the dome on our scale model is about 2. inches. a. What is the diameter of the actual dome? b. bout how man times as tall as our model is the actual building? Solution a. 0 in. 2. in. } 5} 560 ft ft measurement on model measurement on actual building ross roducts ropert Solve for. c The diameter of the actual dome is about 8 feet. b. To simplif a ratio with unlike units, multipl b a conversion factor. 560 ft 560 ft 2 in. } 5} p} in. 0 in. ft c The actual building is 672 times as tall as the model. GUI TI for amples 4 and 5 4. Two cities are 96 miles from each other. The cities are 4 inches apart on a map. ind the scale of the map. 5. WHT I? Your friend has a model of the eunion Tower that is 4 inches tall. What is the diameter of the dome on our friend s model? 366 hapter 6 Similarit

14 6.2 XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s., 3, and 25 5 STNIZ TST TI s. 2, 8, and 24. VOULY op and complete:? is a drawing that has the same shape as the object it represents. 2. WITING Suppose the scale of a model of the iffel Tower is inch : 20 feet. plain how to determine how man times taller the actual tower is than the model. XML on p. 364 for s. 3 0 SONING op and complete the statement. 3. If 8 } 5 3 }, then 8 } 3 5? }?. 4. If } 9 5 } 20, then } 5? }?. 5. If } 6 5 } 5, then 6 } 6 5? }?. 6. If 4 } 3 5 }, then 7 } 3 5? }?. SONING ecide whether the statement is true or false. 7. If 8 } m 5 n } 9, then 8 m } m 5 n 9 } If 5 } 7 5 a } b, then 7 } 5 5 a } b. 9. If } d 5 g 0 d }, then } 5 } If 4 } 5 3 }, then } 5 } 3. 2 g XML 2 on p. 365 for s. 2 OTIS O OOTIONS Use the diagram and the given information to find the unknown length.. Given } 5 }, find. 2. Given XW } XV 5 YW } ZV, find ZV Z Y 24 X 2 W 5 V XMLS 3 and 4 on pp for s. 3 4 SL IGMS In ercises 3 and 4, use the diagram of the field hocke field in which inch 5 50 ards. Use a ruler to approimate the dimension. 3. ind the actual length of the field. 4. ind the actual width of the field. 5. O NLYSIS escribe and correct the error made in the reasoning. If a } 3 5 c } 4, then a 3 } 3 5 c 3 } Use roportions to Solve Geometr roblems 367

15 OTIS O OOTIONS Use the diagram and the given information to find the unknown length. 6. Given } 5 }, find. 7. Given SQ } S 5 TV } TU, find Q S T 3 U 7 V 8. MULTIL HOI If,, z, and q are four different numbers, and the proportion } 5 } z is true, which of the following is false? q } 5 q } z } z 5 } q } 5 z } q } 5 z q } q HLLNG Two number patterns are proportional if there is a nonzero number k such that (a, b, c,...) 5 k(a 2, b 2, c 2,...) 5 ka 2, kb 2, kc 2, Given the relationship (8, 6, 20) 5 k(2, 4, 5), find k. 20. Given that a 5 ka 2, b 5 kb 2, and c 5 kc 2, show that a } a2 5 b } b2 5 c } c2. 2. Given that a 5 ka 2, b 5 kb 2, and c 5 kc 2, show that a b c } a2 b 2 c 2 5 k. OLM SOLVING XML 5 on p. 366 for HITTU basket manufacturer has headquarters in an office building that has the same shape as a basket the sell. a. The bottom of the basket is a rectangle with length 5 inches and width 0 inches. The base of the building is a rectangle with length 92 feet. What is the width of the base of the building? b. bout how man times as long as the bottom of the basket is the base of the building? Longaberger ompan Home Office Newark, Ohio 23. M SL street on a map is 3 inches long. The actual street is mile long. ind the scale of the map. 24. MULTIL HOI model train engine is 2 centimeters long. The actual engine is 8 meters long. What is the scale of the model? 3 cm:2 m cm:.5 m cm:3 m 200 cm:3 m WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

16 M ING The map of a hiking trail has a scale of inch : 3.2 miles. Use a ruler to approimate the actual distance between the two shelters. Meadow View Whispering ines lueberr Hill 25. Meadow View and Whispering ines 26. Whispering ines and lueberr Hill 27. OLLN The photograph shows a particle of goldenrod pollen that has been magnified under a microscope. The scale of the photograph is 900 :. Use a ruler to estimate the width in millimeters of the particle. M SIGN ssume that the wheelchair ramps described each have a slope of }, which is the maimum slope recommended for a wheelchair ramp wheelchair ramp has a 2 foot run. What is its rise? 29. wheelchair ramp rises 4 feet. What is its run? 30. STTISTIS esearchers asked 4887 people to pick a number between and 0. The results are shown in the table below. nswer ercent 4.2% 5.%.4% 0.5% 0.7% nswer ercent 0.0% 27.2% 8.8% 6.0% 6.% a. stimate the number of people who picked the number 3. b. You ask a participant what number she picked. Is the participant more likel to answer 6 or 7? plain. c. onduct this eperiment with our classmates. Make a table in which ou compare the new percentages with the ones given in the original surve. Wh might the be different? LG Use algebra to verif the propert of proportions. 3. ropert ropert ropert Use roportions to Solve Geometr roblems 369

17 SONING Use algebra to eplain wh the propert of proportions is true. 34. If a 2 b } a b 5 c 2 d } c d, then a } b 5 c } d. 35. If a c } b d 5 a 2 c } b 2 d, then a } b 5 c } d. 36. If a } b 5 c } d 5 e } f, then a c e } b d f 5 a } b. (Hint: Let a } b 5 r.) 37. HLLNG When fruit is dehdrated, water is removed from the fruit. The water content in fresh apricots is about 86%. In dehdrated apricots, the water content is about 75%. Suppose 5 kilograms of raw apricots are dehdrated. How man kilograms of water are removed from the fruit? What is the approimate weight of the dehdrated apricots? MIX VIW 38. Over the weekend, laudia drove a total of 405 miles, driving twice as far on Saturda as on Sunda. How far did laudia travel each da? (p. 65) VIW repare for Lesson 6.3 in s Identif all pairs of congruent corresponding parts. Then write another congruence statement for the figures. (p. 225) 39. nxyz > nlmn 40. G > QST X Y N Z L M G S T QUIZ for Lessons Solve the proportion. (p. 356). 0 } 5 5 } 2 2. } } 3 3. } a } } d } 8 op and complete the statement. (p. 364) 5. If 9 } 5 5 } 2, then 9 } 5 5? }?. 6. If } 5 5 } 2, then } 5? }?. 7. If } 8 5 } 2, then 8 } 8 5? }?. 8. If 32 } 5 5 }, then 37 } 5 5? }?. 9. In the diagram, 5 0, is the midpoint of }, and is the geometric mean of and. ind. (p. 364) 370 XT TI for Lesson 6.2, p. 906 ONLIN QUIZ at classzone.com

18 Investigating g Geometr TIVITY Use before Lesson Similar olgons MTILS metric ruler protractor QUSTION When a figure is reduced, how are the corresponding angles related? How are the corresponding lengths related? XLO ompare measures of lengths and angles in two photos ST Measure segments hoto 2 is a reduction of hoto. In each photo, find to the nearest millimeter. Write the ratio of the length of } in hoto to the length of } in hoto 2. 2 ST 2 Measure angles Use a protractor to find the measure of in each photo. Write the ratio of m in hoto to m in hoto 2. ST 3 ind measurements op and complete the table. Use the same units for each measurement. ecord our results in a table. hoto Measurement hoto hoto 2 hoto } hoto 2??? 2?????? m??? m 2??? hoto 2 W ONLUSIONS Use our observations to complete these eercises. Make a conjecture about the relationship between corresponding lengths when a figure is reduced. 2. Make a conjecture about the relationship between corresponding angles when a figure is reduced. 3. Suppose the measure of an angle in hoto 2 is 358. What is the measure of the corresponding angle in hoto? 4. Suppose a segment in hoto 2 is centimeters long. What is the measure of the corresponding segment in hoto? 5. Suppose a segment in hoto is 5 centimeters long. What is the measure of the corresponding segment in hoto 2? 6.3 Use Similar olgons 37

19 6.3 Use Similar olgons efore Now You used proportions to solve geometr problems. You will use proportions to identif similar polgons. Wh? So ou can solve science problems, as in. 34. Ke Vocabular similar polgons scale factor Two polgons are similar polgons if corresponding angles are congruent and corresponding side lengths are proportional. In the diagram below, is similar to GH. You can write is similar to GH as, GH. Notice in the similarit statement that the corresponding vertices are listed in the same order. G orresponding angles >, >, > G, and > H, GH H atios of corresponding sides } 5 } 5 } 5 } G GH H XML Use similarit statements In the diagram, nst, nxyz. a. List all pairs of congruent angles. T Z b. heck that the ratios of corresponding side lengths are equal VOULY In a statement of proportionalit, an pair of ratios forms a true proportion. c. Write the ratios of the corresponding side lengths in a statement of proportionalit. Solution a. > X, S > Y, and T > Z. 20 S X 2 Y b. S } XY 5 20 } } 3 ST }YZ 5 30 } } 3 T } ZX 5 25 } } 3 c. ecause the ratios in part (b) are equal, S } XY 5 ST } YZ 5 T } ZX. GUI TI for ample. Given n JKL, nq, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionalit. 372 hapter 6 Similarit

20 SL TO If two polgons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. In ample, the common ratio of } 5 is the scale factor of nst to nxyz. 3 XML 2 ind the scale factor etermine whether the polgons are similar. If the are, write a similarit statement and find the scale factor of ZYXW to GHJ. Solution G J 2 H Y 25 Z W 5 X ST Identif pairs of congruent angles. rom the diagram, ou can see that Z >, Y > G, and X > H. ngles W and J are right angles, so W > J. So, the corresponding angles are congruent. ST 2 Show that corresponding side lengths are proportional. ZY } 5 } YX } } 5 } XW } }HJ 5 } WZ } } 5 } 20 5 } 5 G 20 4 GH J 6 4 The ratios are equal, so the corresponding side lengths are proportional. c So ZYXW, GHJ. The scale factor of ZYXW to GHJ is 5 } 4. XML 3 Use similar polgons LG In the diagram, n, nmn. ind the value of. Solution The triangles are similar, so the corresponding side lengths are proportional. 9 2 N 2 M 6 20 NOTH WY There are several was to write the proportion. or eample, ou could write } 5 }. M N MN } 5 } N Write proportion. 2 } 5 } 20 Substitute ross roducts ropert 5 5 Solve for. GUI TI for amples 2 and 3 In the diagram,, QST What is the scale factor of QST to? 3. ind the value of. 0 Q T 8 S 6.3 Use Similar olgons 373

21 IMTS The ratios of lengths in similar polgons is the same as the scale factor. Theorem 6. shows this is true for the perimeters of the polgons. THOM or Your Notebook THOM 6. erimeters of Similar olgons If two polgons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. K N L M S ΠIf KLMN, QS, then roof:. 38, p. 379 KL LM MN NK }} 5 } KL 5 } LM 5 } MN 5 } NK. Q Q S S Q Q S S XML 4 ind perimeters of similar figures SWIMMING town is building a new swimming pool. n Olmpic pool is rectangular with length 50 meters and width 25 meters. The new pool will be similar in shape, but onl 40 meters long. a. ind the scale factor of the new pool to an Olmpic pool. b. ind the perimeter of an Olmpic pool and the new pool. NOTH WY nother wa to solve ample 4 is to write the scale factor as the decimal 0.8. Then, multipl the perimeter of the Olmpic pool b the scale factor to get the perimeter of the new pool: 0.8(50) Solution a. ecause the new pool will be similar to an Olmpic pool, the scale factor is the ratio of the lengths, } 40 5 } b. The perimeter of an Olmpic pool is 2(50) 2(25) 5 50 meters. You can use Theorem 6. to find the perimeter of the new pool. } } 5 Use Theorem 6. to write a proportion Multipl each side b 50 and simplif. c The perimeter of the new pool is 20 meters. GUI TI for ample 4 In the diagram,, GHJK. 4. ind the scale factor of GHJK to. 5. ind the value of G 9 2 H 6. ind the perimeter of. K 5 J 374 hapter 6 Similarit

22 SIMILITY N ONGUN Notice that an two congruent figures are also similar. Their scale factor is :. Inn and n, the scale factor is } You can write n, n and n > n. 6 6 VOULY or eample, corresponding lengths in similar triangles include side lengths, altitudes, medians, midsegments, and so on. OSONING LNGTHS You know that perimeters of similar polgons are in the same ratio as corresponding side lengths. You can etend this concept to other segments in polgons. KY ONT orresponding Lengths in Similar olgons or Your Notebook If two polgons are similar, then the ratio of an two corresponding lengths in the polgons is equal to the scale factor of the similar polgons. XML 5 Use a scale factor In the diagram, nt, nxz. ind the length of the altitude } S. Solution T 6 S 6 20 X 8 Y 8 Z irst, find the scale factor of nt to nxz. T } 5 6 } 6 5 } 2 5 } 3 XZ ecause the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, ou can write the following proportion. S } Y 5 3 } 4 Write proportion. S } } 4 Substitute 20 for Y. S 5 5 Multipl each side b 20 and simplif. c The length of the altitude } S is 5. at classzone.com GUI TI for ample 5 7. In the diagram, njkl, ng. ind the length of the median } KM. K G 40 H 35 J 48 M L 6.3 Use Similar olgons 375

23 6.3 XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s. 3, 7, and 3 5 STNIZ TST TI s. 2, 6, 8, 27, 28, 35, 36, and 37 5 MULTIL SNTTIONS. 33. VOULY op and complete: Two polgons are similar if corresponding angles are? and corresponding side lengths are?. 2. WITING If two polgons are congruent, must the be similar? If two polgons are similar, must the be congruent? plain. XML on p. 372 for s. 3 6 USING SIMILITY List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionalit. 3. n, nlmn 4. G, QS 5. HJKL, WXYZ L H N M G S L X K J Y Z W 6. MULTIL HOI Triangles and are similar. Which statement is not correct? } 5 } } 5 } } 5 } } 5 } XMLS 2 and 3 on p. 373 for s. 7 0 TMINING SIMILITY etermine whether the polgons are similar. If the are, write a similarit statement and find the scale factor S 48 Z W V U 8 U 64 T Y 24 X 5 T USING SIMIL OLYGONS In the diagram, JKLM, GH. J 9. ind the scale factor of JKLM to GH. 0. ind the values of,, and z. H 3 G M XML 4 on p. 374 for s. 3. ind the perimeter of each polgon. 2. IMT Two similar O SL signs have a scale factor of 5 : 3. The large sign s perimeter is 60 inches. ind the small sign s perimeter. z8 8 K L 3. O NLYSIS The triangles are similar. escribe and correct the error in finding the perimeter of Triangle erimeter of hapter 6 Similarit

24 SONING re the polgons alwas, sometimes, or never similar? 4. Two isosceles triangles 5. Two equilateral triangles 6. right triangle and an isosceles triangle 7. scalene triangle and an isosceles triangle 8. SHOT SONS The scale factor of igure to igure is :. What is the scale factor of igure to igure? plain our reasoning. SIMIL TINGLS Identif the tpe of special segment shown in blue, and find the value of the variable XML 5 on p. 375 for s USING SL TO Triangles NQ and ST are similar. The side lengths of nnq are 6 inches, 8 inches, and 0 inches, and the length of an altitude is 4.8 inches. The shortest side of nst is 8 inches long. 2. ind the lengths of the other two sides of nst. 22. ind the length of the corresponding altitude in nst. USING SIMIL TINGLS In the diagram, n, n. 23. ind the scale factor of n to n. 24. ind the unknown side lengths in both triangles ind the length of the altitude shown in n. 26. ind and compare the areas of both triangles SHOT SONS Suppose ou are told that nq, nxyz and that the etended ratio of the angle measures in nq is : 30:3. o ou need to know anthing about nxyz to be able to write its etended ratio of angle measures? plain our reasoning. 28. MULTIL HOI The lengths of the legs of right triangle are 3 feet and 4 feet. The shortest side of nuvw is 4.5 feet and nuvw, n. How long is the hpotenuse of nuvw?.5 ft 5ft 6ft 7.5 ft 29. HLLNG op the figure at the right and divide it into two similar figures. 30. SONING Is similarit refleive? smmetric? transitive? Give eamples to support our answers. 6.3 Use Similar olgons 377

25 OLM SOLVING XML 2 on p. 373 for s TNNIS In table tennis, the table is a rectangle 9 feet long and 5 feet wide. tennis court is a rectangle 78 feet long and 36 feet wide. re the two surfaces similar? plain. If so, find the scale factor of the tennis court to the table. 32. IGITL OJTO You are preparing a computer presentation to be digitall projected onto the wall of our classroom. Your computer screen is 3.25 inches wide and 0.6 inches high. The projected image on the wall is 53 inches wide and 42.4 inches high. re the two shapes similar? If so, find the scale factor of the computer screen to the projected image. 33. MULTIL SNTTIONS Use the similar figures shown. The scale factor of igure to igure 2 is 7 : 0. a. Making a Table op and complete the table igure 3.5???? 8 igure igure igure 2 b. rawing a Graph Graph the data in the table. Let represent the length of a side in igure and let represent the length of the corresponding side in igure 2. Is the relationship linear? c. Writing an quation Write an equation that relates and. What is its slope? How is the slope related to the scale factor? 34. MULTI-ST OLM uring a total eclipse of the sun, the moon is directl in line with the sun and blocks the sun s ras. The distance between arth and the moon is 240,000 miles, the distance between arth and the sun is 93,000,000 miles, and the radius of the sun is 432,500 miles. a. op the diagram and label the known distances. b. In the diagram, n, n. Use this fact to eplain a total eclipse of the sun. c. stimate the radius of the moon WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI 5 MULTIL SNTTIONS

26 35. SHOT SONS rectangular image is enlarged on each side b the same amount. The angles remain unchanged. an the larger image be similar to the original? plain our reasoning, and give an eample to support our answer. l w l a w a 36. SHOT SONS How are the areas of similar rectangles related to the scale factor? Use eamples to justif our reasoning. 37. XTN SONS The equations of two lines in the coordinate plane are 5 } 4 4 and 5 } a. plain wh the two lines are parallel. b. Show that O > O, O > O, and O > O. O c. ind the coordinates of points,,, and. ind the lengths of the sides of n O and no. d. Show that n O, no. 38. OVING THOM 6. rove the erimeters of Similar olgons Theorem for similar rectangles. Include a diagram in our proof. 39. HLLNG In the diagram, QS is a square, and LMS, LMQ. ind the eact value of. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. Show that the similar rectangles in the diagram are golden rectangles. S L M VIW repare for Lesson 6.4 in s MIX VIW Given (, ), (3, 2), (2, 4), and, } 7 2 2, determine whether the following lines are parallel, perpendicular, or neither. (p. 7) 40. ] and ] 4. ] and ] 42. ind the measure of the eterior angle shown. (p. 27) ] and ] 43. (3 2 50) (2 20) (3 8)8 258 [6( 2 2)]8 op and complete the statement with <, >, or 5. (p. 335) 46. S? TU 47. G? H 48. WX? YX U V S T H G W 488 X Z 458 Y XT TI for Lesson 6.3, p. 906 ONLIN QUIZ at classzone.com 379

27 MIX VIW of roblem Solving STT TST TI classzone.com Lessons MULTI-ST OLM In the diagram, nlmn, nqs. M SHOT SONS Kell bought a 3- scale model of the Tower ridge in London, ngland. The towers of the model are 9 inches tall. The towers of the actual bridge are 206 feet tall, and there are two walkwas that are 40 feet high. L 6 N S a. ind the scale factor of nlmn to nqs. Then find the values of and. b. ind the perimeters of nlmn and nqs. c. ind the areas of nlmn and nqs. d. ompare the ratio of the perimeters to the ratio of the areas of nlmn to nqs. What do ou notice? 2. GI NSW In the diagram, : is 3:8. ind ON-N nuvw is a right triangle with side lengths of 3 cm, 4 cm, and 5 cm. raw and label nuvw. Then draw a triangle similar to nuvw and label its side lengths. What scale factor did ou use? 4. MULTI-ST OLM Kell is going on a trip to ngland. She takes 600 U.S. dollars with her. a. In ngland, she echanges her U.S. dollars for ritish pounds. uring her sta, Kell spends 50 pounds. How man ritish pounds does she have left? b. When she returns home, she echanges her mone back to U.S. dollars. How man U.S. dollars does she have at the end of her trip? a. pproimate the height of the walkwas on the model. b. bout how man times as tall as the model is the actual structure? 6. GI NSW In the diagram, n, n. The scale factor of n to n is 3:5. ind. 7. XTN SONS In the United States, 4634 million pounds of apples were consumed in The population of the United States in that ear was 290 million. a. ivide the total number of apples consumed b the population to find the per capita consumption. b. bout how man pounds of apples would a famil of four have consumed in one ear? in one month? c. medium apple weighs about 5 ounces. stimate how man apples a famil of four would have consumed in one month. d. Is it reasonable to assume that a famil of four would have eaten that man apples? What other factors could affect the per capita consumption? plain hapter 6 Similarit

28 6.4 rove Triangles Similar b efore You used the S ongruence Theorem. Now You will use the Similarit ostulate. Wh? So ou can use similar triangles to understand aerial photograph, as in. 34. Ke Vocabular similar polgons, p. 372 TIVITY NGLS N SIMIL TINGLS QUSTION What can ou conclude about two triangles if ou know two pairs of corresponding angles are congruent? Materials: protractor metric ruler ST raw ng so that m and m G ST 2 raw nst so that m and m T 5 508, and nst is not congruent to ng. ST 3 alculate m and m S using the Triangle Sum Theorem. Use a protractor to check that our results are true. ST 4 Measure and record the side lengths of both triangles. Use a metric ruler. 40 S 50 40º 50º G T W ONLUSIONS. re the triangles similar? plain our reasoning. 2. epeat the steps above using different angle measures. Make a conjecture about two triangles with two pairs of congruent corresponding angles. TINGL SIMILITY The ctivit suggests that two triangles are similar if two pairs of corresponding angles are congruent. In other words, ou do not need to know the measures of the sides or the third pair of angles. OSTULT or Your Notebook OSTULT 22 ngle-ngle () Similarit ostulate If two angles of one triangle are congruent K to two angles of another triangle, then the two triangles are similar. J L X njkl, nxyz Y Z 6.4 rove Triangles Similar b 38

29 XML Use the Similarit ostulate W IGMS H G K Use colored pencils to show congruent angles. This will help ou write similarit statements. etermine whether the triangles are H similar. If the are, write a similarit statement. plain our reasoning. G Solution ecause the are both right angles, and G are congruent. the Triangle Sum Theorem, m 5 808, so m Therefore, and H are congruent. c So, n, nkgh b the Similarit ostulate. K XML 2 Show that triangles are similar Show that the two triangles are similar. a. n and n b. nsv and nuvt T 528 S V 528 U Solution a. You ma find it helpful to redraw the triangles separatel. ecause m and m both equal 528, >. the efleive ropert, >. c So, n, n b the Similarit ostulate. b. You know SV > UVT b the Vertical ngles ongruence Theorem. The diagram shows } S i } UT so S > U b the lternate Interior ngles Theorem. c So, nsv, nuvt b the Similarit ostulate. S V T U GUI TI for amples and 2 Show that the triangles are similar. Write a similarit statement.. ngh and nqs 2. n and n G H S SONING Suppose in ample 2, part (b), } S i } TU. ould the triangles still be similar? plain. 382 hapter 6 Similarit

30 INIT MSUMNT In Lesson 4.6, ou learned a wa to use congruent triangles to find measurements indirectl. nother useful wa to find measurements indirectl is b using similar triangles. XML 3 Standardized Test ractice LIMINT HOIS Notice that the woman s height is greater than her shadow s length. So the flagpole must be taller than its shadow s length. liminate choices and. flagpole casts a shadow that is 50 feet long. t the same time, a woman standing nearb who is five feet four inches tall casts a shadow that is 40 inches long. How tall is the flagpole to the nearest foot? 2 feet 80 feet 40 feet 40 feet Solution The flagpole and the woman form sides of two right triangles with the ground, as shown below. The sun s ras hit the flagpole and the woman at the same angle. You have two pairs of congruent angles, so the triangles are similar b the Similarit ostulate. 5 ft 4 in. ft 40 in. 50 ft You can use a proportion to find the height. Write 5 feet 4 inches as 64 inches so that ou can form two ratios of feet to inches. ft 50 ft } 5 } Write proportion of side lengths. 64 in. 40 in (50) ross roducts ropert 5 80 Solve for. c The flagpole is 80 feet tall. The correct answer is. GUI TI for ample 3 4. WHT I? child who is 58 inches tall is standing net to the woman in ample 3. How long is the child s shadow? 5. You are standing in our backard, and ou measure the lengths of the shadows cast b both ou and a tree. Write a proportion showing how ou could find the height of the tree. 6.4 rove Triangles Similar b 383

31 6.4 XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s. 9, 3, and 33 5 STNIZ TST TI s. 2, 6, 8, 9, 20, 33, and 38. VOULY op and complete: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are?. 2. WITING an ou assume that corresponding sides and corresponding angles of an two similar triangles are congruent? plain. XML on p. 382 for s. 3 SONING Use the diagram to complete the statement. 3. n,? }? 5? } 2 6. } 5 } 5 }???? } }? ? 8. 5? 25 5 SIMILITY OSTULT In ercises 9 4, determine whether the triangles are similar. If the are, write a similarit statement. 9. K G H J L 0. N M 458 Y X 458 Z. 358 V 858 U 658 S 358 T XML 2 on p. 382 for s W Y X 508 Z U 4. L M N 5. O NLYSIS plain wh the student s similarit statement is incorrect., GH b Similarit ostulate G H 6. MULTIL HOI What is the value of p? annot be determined p 384 hapter 6 Similarit

32 7. O NLYSIS student uses the proportion 4 } 5 } 5 to find the value of in the figure. plain 6 wh this proportion is incorrect and write a correct proportion ON-N MTH In ercises 8 and 9, make a sketch that can be used to show that the statement is false. 8. If two pairs of sides of two triangles are congruent, then the triangles are similar. 9. If the ratios of two pairs of sides of two triangles are proportional, then the triangles are similar. 20. MULTIL HOI In the figure at the right, find the length of }. } 35 3 } 37 5 } 20 3 } LG ind coordinates for point so that n S n. 2. (0, 0), (0, 4), (8, 0), (0, 5), (, ) 22. (0, 0), (0, 3), (4, 0), (0, 7), (, ) 23. (0, 0), (0, ), (6, 0), (0, 4), (, ) 24. (0, 0), (0, 6), (3, 0), (0, 9), (, ) 25. MULTI-ST OLM In the diagram, ] i ], 5 6, 5 8, 5 5, and 5 0. a. op the diagram and mark all given information. b. List two pairs of congruent angles in the diagram. c. Name a pair of similar triangles and write a similarit statement. d. ind and. SONING In ercises 26 29, is it possible for njkl and nxyz to be similar? plain wh or wh not. 26. m J 5 78, m K 5 528, m X 5 78, and m Z njkl is a right triangle and m X m Y m J and m Y m J m K and m Y m Z HLLNG If T 5, Q 5 3, and S 5 8 } 3, find S in terms of. plain our reasoning. S T 6.4 rove Triangles Similar b 385

33 OLM SOLVING XML 3 on p. 383 for s I HOKY n air hocke plaer returns the puck to his opponent b bouncing the puck off the wall of the table as shown. rom phsics, the angles that the path of the puck makes with the wall are congruent. What is the distance d between the puck and the wall when the opponent returns it? puck 20 in. 26 in. 66 in. d wall 32. LKS You can measure the width of the lake using a surveing technique, as shown in the diagram. V 04 m a. What postulate or theorem can ou use to show that the triangles are similar? b. ind the width of the lake, WX. c. If XY 5 0 meters, find VX. X 6 m Z 8 m Y W 33. SHOT SONS plain wh all equilateral triangles are similar. Include sketches in our answer. 34. IL HOTOGHY Low-level aerial photos can be taken using a remote-controlled camera suspended from a blimp. You want to take an aerial photo that covers a ground distance g of 50 meters. Use the proportion } f 5 } n to estimate h g the altitude h that the blimp should fl at to take the photo. In the proportion, use f 5 8 centimeters and n 5 3 centimeters. These two variables are determined b the tpe of camera used. 35. OO Use the given information to draw a sketch. Then write a proof. GIVN c nstu, nq oint V lies on } TU so that } SV bisects TSU. oint N lies on } Q so that } N bisects Q. OV c SV } N 5 ST } Q 36. OO rove that if an acute angle in one right triangle is congruent to an acute angle in another right triangle, then the triangles are similar WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

34 37. THNOLOGY Use a graphing calculator or computer. a. raw n. raw } through two sides of the triangle, parallel to the third side. b. Measure and. Measure and. What do ou notice? c. What does a postulate in this lesson tell ou about n and n? d. Measure all the sides. Show that corresponding side lengths are proportional. e. Move verte to form new triangles. How do our measurements in parts (b) and (d) change? re the new triangles still similar? plain. 38. XTN SONS plain how ou could use similar triangles to show that an two points on a line can be used to calculate its slope. 39. OSONING LNGTHS Without using the orresponding Lengths ropert on page 375, prove that the ratio of two corresponding angle bisectors in similar triangles is equal to the scale factor. 40. HLLNG rove that if the lengths of two sides of a triangle are a and b respectivel, then the lengths of the corresponding altitudes to those sides are in the ratio } b. a MIX VIW VIW repare for Lesson 6.5 in s In ercises 4 44, use the diagram. 4. Name three pairs of corresponding angles. (p. 47) 42. Name two pairs of alternate interior angles. (p. 47) 43. Name two pairs of alternate eterior angles. (p. 47) ind m m 7. (p. 54) 45. ONGUN plain wh n > n. (p. 240) Simplif the ratio. (p. 356) } } : :28 XT TI for Lesson 6.4, p. 907 ONLIN QUIZ at classzone.com 387

35 6.5 rove Triangles Similar b SSS and SS efore You used the Similarit ostulate to prove triangles similar. Now You will use the SSS and SS Similarit Theorems. Wh? So ou can show that triangles are similar, as in. 28. Ke Vocabular ratio, p. 356 proportion, p. 358 similar polgons, p. 372 In addition to using congruent corresponding angles to show that two triangles are similar, ou can use proportional corresponding side lengths. THOM THOM 6.2 Side-Side-Side (SSS) Similarit Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar. If } 5 } 5 }, then n, nst. S ST T roof: p. 389 or Your Notebook S T XML Use the SSS Similarit Theorem Is either n or nghj similar to n? J 6 2 H G LY THOMS When using the SSS Similarit Theorem, compare the shortest sides, the longest sides, and then the remaining sides. Solution ompare n and n b finding ratios of corresponding side lengths. Shortest sides Longest sides emaining sides } 5 } } } 5 } } } 5 } 2 5 } c ll of the ratios are equal, so n, n. ompare n and nghj b finding ratios of corresponding side lengths. Shortest sides Longest sides emaining sides } GH 5 8 } 8 5 } JG 5 6 } 6 5 } HJ 5 2 } } 5 c The ratios are not all equal, so n and nghj are not similar. 388 hapter 6 Similarit

36 OO SSS Similarit Theorem GIVN c S } JK 5 ST } KL 5 T } LJ K S US N UXILIY LIN The arallel ostulate allows ou to draw an auiliar line ] Q in nst. There is onl one line through point parallel to ] T, so ou are able to draw it. OV c nst, njkl J Locate on } S so that S 5 JK. raw } Q so that } Q i } T. Then nst, nsq b the Similarit ostulate, and } S 5 } ST 5 } T. S SQ Q You can use the given proportion and the fact that S 5 JK to deduce that SQ 5 KL and Q 5 LJ. the SSS ongruence ostulate, it follows that n SQ > n JKL. inall, use the definition of congruent triangles and the Similarit ostulate to conclude that n ST, n JKL. L ΠT XML 2 Use the SSS Similarit Theorem LG ind the value of that makes n, n HOOS MTHO You can use either } 5 } or } 5 } in Step. Solution ST ind the value of that makes corresponding side lengths proportional. 4 } } 8 Write proportion. 4 p 8 5 2( 2 ) ross roducts ropert Simplif. 3( ) 75 Solve for. ST 2 heck that the side lengths are proportional when ( ) 5 24 } 0 } 4 }2 5 6 } 8 } 0 } 4 }2 5 8 } 24 c When 5 7, the triangles are similar b the SSS Similarit Theorem. GUI TI for amples and 2. Which of the three triangles are similar? Write a similarit statement. 2. The shortest side of a triangle similar to nst is 2 units long. ind the other side lengths of the triangle. 20 L M 24 N T S 39 X Y Z 6.5 rove Triangles Similar b SSS and SS 389

37 THOM or Your Notebook THOM 6.3 Side-ngle-Side (SS) Similarit Theorem If an angle of one triangle is congruent to an angle X of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Z If X > M and } ZX 5 } XY, then n XYZ, n MN. M MN roof:. 37, p. 395 M Y N XML 3 Use the SS Similarit Theorem LN-TO SHLT You are building a lean-to shelter starting from a tree branch, as shown. an ou construct the right end so it is similar to the left end using the angle measure and lengths shown? 53º 6 ft G 0 ft H 53º 9 ft 5 ft Solution oth m and m equal 538, so >. Net, compare the ratios of the lengths of the sides that include and. Shorter sides } G 5 9 } } 2 Longer sides } H 5 5 } } 2 The lengths of the sides that include and are proportional. c So, b the SS Similarit Theorem, n, ngh. Yes, ou can make the right end similar to the left end of the shelter. ONT SUMMY or Your Notebook Triangle Similarit ostulate and Theorems Similarit ostulate SSS Similarit Theorem SS Similarit Theorem If > and >, then n, n. If } 5 } 5 }, then n, n. If > and } 5 }, then n, n. 390 hapter 6 Similarit

38 XML 4 hoose a method VISUL SONING To identif corresponding parts, redraw the triangles so that the corresponding parts have the same orientation Tell what method ou would use to show that the triangles are similar. Solution ind the ratios of the lengths of the corresponding sides. Shorter sides } 5 9 } } 5 Longer sides } 5 8 } } 5 The corresponding side lengths are proportional. The included angles and are congruent because the are vertical angles. So, n, n b the SS Similarit Theorem at classzone.com GUI TI for amples 3 and 4 plain how to show that the indicated triangles are similar. 3. nst, nnq 4. nxzw, n YZX S X T N 2 W 6 Z 9 Y 6.5 XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s. 3, 7, and 3 5 STNIZ TST TI s. 2, 4, 32, 34, and 36. VOULY You plan to prove that n is similar to nxq b the SSS Similarit Theorem. op and complete the proportion that is needed to use this theorem: } 5 }? 5 }.? XQ? 2. WITING If ou know two triangles are similar b the SS Similarit Theorem, what additional piece(s) of information would ou need to know to show that the triangles are congruent? XMLS and 2 on pp for s. 3 6 SSS SIMILITY THOM Verif that n, n. ind the scale factor of n to n. 3. n: 5 8, 5 5, n: 5 0, 5 6, 5 20 n: 5 2, 5 0, 5 8 n: 5 25, 5 40, rove Triangles Similar b SSS and SS 39

39 5. SSS SIMILITY THOM Is either njkl or nst similar to n? 8 K 7 L S 4 T J 6. SSS SIMILITY THOM Is either n JKL or nst similar to n? L T K 20 J 2 S XML 3 on p. 390 for s. 7 9 SS SIMILITY THOM etermine whether the two triangles are similar. If the are similar, write a similarit statement and find the scale factor of Triangle to Triangle Y 6 X W 8. 8 S 28 8 T L LG ind the value of n that makes nq, nxyz when Q 5 4, Q 5 5, XY 5 4(n ), YZ 5 7n 2, and Q > Y. Include a sketch. K 24 J XML 4 on p. 39 for s. 0 2 SHOWING SIMILITY Show that the triangles are similar and write a similarit statement. plain our reasoning. 0. H 5 G J 5.5 K X Y G Z J O NLYSIS escribe and correct the student s error in writing the similarit statement. n, n Q b SS Similarit Theorem Q 4. MULTIL HOI In the diagram, } MN 5 } M. M MQ Which of the statements must be true? > 2 } Q i } N > 4 nmn, nmq 2 N M WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

40 WING TINGLS Sketch the triangles using the given description. plain whether the two triangles can be similar. 5. In nxyz, m X and m Y In nlmn, m M and m N In nst, S 5 20, ST 5 32, and m S In ngh, GH 5 30, H 5 48, and m H The side lengths of n are 24, 8, and 54, and the side lengths of n are 5, 25, and 7. INING MSUS In ercises 8 23, use the diagram to cop and complete the statements. L 2 28 S M 8. m NQ 5? 9. m QN 5? 20. m NQ 5? 2. N5? N Q 5? 23. NM 5? SIMIL TINGLS In the diagram at the right, name the three pairs of triangles that are similar. 8 HLLNG In the figure at the right, n, nvwx. 25. ind the scale factor of nvwx to n. 26. ind the ratio of the area of nvwx to the area of n. V 5 30 X 45 Y W Make a conjecture about the relationship between the scale factor in ercise 25 and the ratio in ercise 26. Justif our conjecture. 2 OLM SOLVING 28. NT Which postulate or theorem could ou use to show that the three triangles that make up the racecar window net are similar? plain. }G i }, } i } XML on p. 388 for STIN GLSS ertain sections of stained glass are sold in triangular beveled pieces. Which of the three beveled pieces, if an, are similar? 3 in. 5 in. 3 in. 4 in. 7 in. 4 in in. 3 in. 3 in. 6.5 rove Triangles Similar b SSS and SS 393

41 SHULO In the portion of the shuffleboard court shown, } 5 }. 30. What additional piece of information do ou need in order to show that n, n using the SSS Similarit Theorem? 3. What additional piece of information do ou need in order to show that n, n using the SS Similarit Theorem? 32. ON-N MTH Use a diagram to show wh there is no Side-Side-ngle Similarit ostulate. XML 4 on p. 39 for MULTI-ST OLM ub is standing in her back ard and she decides to estimate the height of a tree. She stands so that the tip of her shadow coincides with the tip of the tree s shadow, as shown. ub is 66 inches tall. The distance from the tree to ub is 95 feet and the distance between the tip of the shadows and ub is 7 feet. a. What postulate or theorem can ou use to show that the triangles in the diagram are similar? b. bout how tall is the tree, to the nearest foot? c. What If? urtis is 75 inches tall. t a different time of da, he stands so that the tip of his shadow and the tip of the tree s shadow coincide, as described above. His shadow is 6 feet long. How far is urtis from the tree? at classzone.com 34. XTN SONS Suppose ou are given two right triangles with one pair of corresponding legs and the pair of corresponding hpotenuses having the same length ratios. a. The lengths of the given pair of corresponding legs are 6 and 8, and the lengths of the hpotenuses are 0 and 30. Use the thagorean Theorem to solve for the lengths of the other pair of corresponding legs. raw a diagram. b. Write the ratio of the lengths of the second pair of corresponding legs. c. re these triangles similar? oes this suggest a Hpotenuse-Leg Similarit Theorem for right triangles? 35. OO Given that n is a right triangle and,, and are midpoints, prove that m WITING an two triangles have all pairs of corresponding angles in proportion? plain WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

42 37. OVING THOM 6.3 Write a paragraph proof of the SS Similarit Theorem. GIVN c >, } 5 } OV c n, n G H 38. HLLNG portion of a water slide in an amusement park is shown. ind the length of }. (Note: The posts form right angles with the ground.) MIX VIW VIW repare for Lesson 6.6 in s ind the slope of the line that passes through the given points. (p. 7) 39. (0, 28), (4, 6) 40. (22, 29), (, 23) 4. (23, 9), (7, 2) 42. State the postulate or theorem ou would use to prove the triangles congruent. Then write a congruence statement. (p. 249) ind the value of. 43. } is a midsegment of n. (p. 295) 44. GK } GH 5 JK } H (p. 364) 5 J H T 9 S G 3 K 5 QUIZ for Lessons In the diagram,, KLMN. (p. 372). ind the scale factor of to KLMN. 2. ind the values of,, and z. 3. ind the perimeter of each polgon K 36 z8 L 0 N M etermine whether the triangles are similar. If the are similar, write a similarit statement. (pp. 38, 388) 4. W Z X 6. L M Y 2 N S 35 H G 428 J XT TI for Lesson 6.5, p. 907 ONLIN QUIZ at classzone.com 395

43 Investigating g Geometr TIVITY 6.6 Investigate roportionalit MTILS graphing calculator or computer Use before Lesson 6.6 classzone.com Kestrokes QUSTION How can ou use geometr drawing software to compare segment lengths in triangles? XLO onstruct a line parallel to a triangle s third side ST raw a triangle raw a triangle. Label the vertices,, and. raw a point on }. Label the point. ST 2 raw a parallel line raw a line through that is parallel to }. Label the intersection of the line and } as point. ST 3 Measure segments Measure }, }, }, and }. alculate the ratios } and }. ST 4 ompare ratios Move one or more of the triangle s vertices to change its shape. ompare the ratios from Step 3 as the shape changes. Save as XLO. XLO 2 onstruct an angle bisector of a triangle ST raw a triangle raw a triangle. Label the vertices, Q, and. raw the angle bisector of Q. Label the intersection of the angle bisector and } Q as point. ST 2 Measure segments Measure }, }, } Q, and } Q. alculate the ratios } and }. Q Q Q ST 3 ompare ratios Move one or more of the triangle s vertices to change its shape. ompare the ratios from Step 3. Save as XLO2. W ONLUSIONS Use our observations to complete these eercises. Make a conjecture about the ratios of the lengths of the segments formed when two sides of a triangle are cut b a line parallel to the triangle s third side. 2. Make a conjecture about how the ratio of the lengths of two sides of a triangle is related to the ratio of the lengths of the segments formed when an angle bisector is drawn to the third side. 396 hapter 6 Similarit

44 6.6 Use roportionalit Theorems efore You used proportions with similar triangles. Now You will use proportions with a triangle or parallel lines. Wh? So ou can use perspective drawings, as in. 28. Ke Vocabular corresponding angles, p. 47 ratio, p. 356 proportion, p. 358 The Midsegment Theorem, which ou learned on page 295, is a special case of the Triangle roportionalit Theorem and its converse. THOMS THOM 6.4 Triangle roportionalit Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionall. roof:. 22, p. 402 S or Your Notebook T U If } TU i } QS, then T } TQ 5 U } US. THOM 6.5 onverse of the Triangle roportionalit Theorem If a line divides two sides of a triangle proportionall, then it is parallel to the third side. roof:. 26, p. 402 S T U If T } TQ 5 U } US, then } TU i } QS. XML ind the length of a segment In the diagram, } QS i } UT, S 5 4, ST 5 6, and QU 5 9. What is the length of } Q? U 9 4 S 6 Solution T Q } QU 5 S } ST Triangle roportionalit Theorem Q } } 6 Substitute. Q 5 6 Multipl each side b 9 and simplif. 6.6 Use roportionalit Theorems 397

45 SONING Theorems 6.4 and 6.5 also tell ou that if the lines are not parallel, then the proportion is not true, and vice-versa. So if } TU i } QS, then T } TQ Þ U } US. lso, if T } TQ Þ U } US, then } TU i } QS. XML 2 Solve a real-world problem SHOK On the shoerack shown, 5 33 cm, 5 27 cm, 5 44 cm, and 5 25 cm. plain wh the gra shelf is not parallel to the floor. Solution ind and simplif the ratios of lengths determined b the shoerack. } 5 44 } } 5 } 27 5 } c ecause 44 } 25 Þ 9 }, } is not parallel to }. So, the shelf is not parallel to the floor. GUI TI for amples and 2. ind the length of } YZ. 2. etermine whether } S i } Q. V 35 W 44 Z Y 36 X N S THOMS or Your Notebook THOM 6.6 If three parallel lines intersect two transversals, then the divide the transversals proportionall. r s U W Y V X Z t l m roof:. 23, p. 402 UW } WY 5 VX } XZ THOM 6.7 If a ra bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. roof:. 27, p. 403 } 5 } 398 hapter 6 Similarit

46 XML 3 Use Theorem 6.6 ITY TVL In the diagram,, 2, and 3 are all congruent and G 5 20 ards, 5 50 ards, and ards. ind the distance H between Main Street and South Main Street. 20 G Main St Second St. NOTH WY or alternative methods for solving the problem in ample 3, turn to page 404 for the roblem Solving Workshop. Solution orresponding angles are congruent, so ], ] G, and ] H are parallel. Use Theorem 6.6. HG } G 5 } arallel lines divide transversals proportionall. HG G } 5} ropert of proportions (ropert 4) G H South Main St. H } 5} Substitute H } 5 } 450 Simplif H Multipl each side b 20 and simplif. c The distance between Main Street and South Main Street is 360 ards. XML 4 Use Theorem 6.7 In the diagram, Q > S. Use the given side lengths to find the length of } S. 7 Solution ecause ] is an angle bisector of QS, ou can appl Theorem 6.7. Let S 5. Then Q Q } S 5 Q } S ngle bisector divides opposite side proportionall. S 5 2 } 5 7 } 3 Substitute ross roducts ropert Solve for. GUI TI for amples 3 and 4 ind the length of } Use roportionalit Theorems 399

47 6.6 XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s. 5, 9, and 2 5 STNIZ TST TI s. 2, 8, 3, 25, and 28. VOULY State the Triangle roportionalit Theorem. raw a diagram. 2. WITING ompare the Midsegment Theorem (see page 295) and the Triangle roportionalit Theorem. How are the related? XML on p. 397 for s. 3 4 INING TH LNGTH O SGMNT ind the length of } XML 2 on p. 398 for s. 5 7 SONING Use the given information to determine whether } KM i } JN. plain our reasoning. 5. K 5 J 8 L 6. 2 M 7.5 N L 8 J 0 K 24 M 5 N 7. J 22.5 K 25 L N 8 M 20 XML 3 on p. 399 for MULTIL HOI or the figure at the right, which statement is not necessaril true? } Q 5 } UT } TS 5 } Q Q TS UT Q Q } S 5 TS } S Q } 5 UT } US U T S XML 4 on p. 399 for s. 9 2 LG ind the value of the variable z O NLYSIS student begins to solve for the length of } as shown. escribe and correct the student s error } 5 0 } }6 5} hapter 6 Similarit

48 3. MULTIL HOI ind the value of. } LG ind the value of the variable p 6 q INING SGMNT LNGTHS Use the diagram to find the value of each variable. 6. a b 2.5 e c d f a 3 6 d 6 6 b c O NLYSIS student claims that 5 using the method shown. escribe and correct the student s error. Theorem 6.7, } 5 }. ecause 5, it follows that ONSTUTION ollow the instructions for constructing a line segment that is divided into four equal parts. a. raw a line segment that is about 3 inches long, and label its endpoints and. hoose an point not on }. raw ]. b. Using an length, place the compass point at and make an arc intersecting ] at. Using the same compass setting, make additional arcs on ]. Label the points,, and G so that G. c. raw } G. onstruct a line parallel to } G through. ontinue constructing parallel lines and label the points as shown. plain wh J 5 JK 5 KL 5 L. G J K L G 20. HLLNG Given segments with lengths r, s, and t, construct a segment of length, such that } r 5 } t. s r s t 6.6 Use roportionalit Theorems 40

49 OLM SOLVING 2. ITY M On the map below, Idaho venue bisects the angle between Universit venue and Walter Street. To the nearest ard, what is the distance along Universit venue from 2th Street to Washington Street? 22. OVING THOM 6.4 rove the Triangle roportionalit Theorem. GIVN c } QS i } TU OV c QT } T 5 SU } U S T U 23. OVING THOM 6.6 Use the diagram with the auiliar line drawn to write a paragraph proof of Theorem 6.6. GIVN c k i k 2, k 2 i k 3 OV c } 5 } t t 2 auiliar line k k 2 k MULTI-ST OLM The real estate term lake frontage refers to the distance along the edge of a piece of propert that touches a lake. a. ind the lake frontage (to the nearest tenth of a ard) for each lot shown. b. In general, the more lake frontage a lot has, 74 d the higher its selling price. Which of the lots should be listed for the highest price? c. Suppose that lot prices are in the same ratio as lake frontages. If the least epensive lot is $00,000, what are the prices of the other lots? plain our reasoning. Lot 48 d Lot 55 d Lakeshore rive Lot 6 d lake 25. SHOT SONS Sketch an isosceles triangle. raw a ra that bisects the angle opposite the base. This ra divides the base into two segments. Theorem 6.7, the ratio of the legs is proportional to the ratio of these two segments. plain wh this ratio is : for an isosceles triangle. 26. LN O OO Use the diagram given for the proof of Theorem 6.4 in ercise 22 to write a plan for proving Theorem 6.5, the Triangle roportionalit onverse WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

50 27. OVING THOM 6.7 Use the diagram with the auiliar lines drawn to write a paragraph proof of Theorem 6.7. Y GIVN c YXW > WXZ OV c YW } WZ 5 XY } XZ X auiliar lines Z W 28. XTN SONS In perspective drawing, lines that are parallel in real life must meet at a vanishing point on the horizon. To make the train cars in the drawing appear equal in length, the are drawn so that the lines connecting the opposite corners of each car are parallel. a. Use the dimensions given and the red parallel lines to find the length of the bottom edge of the drawing of ar 2. b. What other set of parallel lines eist in the figure? plain how these can be used to form a set of similar triangles. c. ind the length of the top edge of the drawing of ar HLLNG rove eva s Theorem: If is an point inside n, then } Y p } X p } Z 5. (Hint: raw Y X Z N Z X M lines parallel to } Y through and. ppl Theorem 6.4 to n M. Show that n N, nm, nxm, nx, and nz, n ZN.) Y MIX VIW VIW repare for Lesson 6.7 in s erform the following operations. Then simplif. 30. (23) p 7 } 2 (p. 869) 3. 4 } 3 p } 2 (p. 869) } (p. 87) } (p. 87) escribe the translation in words and write the coordinate rule for the translation. (p. 272) XT TI for Lesson 6.6, p. 907 ONLIN QUIZ at classzone.com 403

51 LSSON 6.6 Using LTNTIV MTHOS nother Wa to Solve ample 3, page 399 MULTIL SNTTIONS In Lesson 6.6, ou used proportionalit theorems to find lengths of segments formed when transversals intersect two or more parallel lines. Now, ou will learn two different was to solve ample 3 on page 399. OLM ITY TVL In the diagram,, 2, and 3 are all congruent and G 5 20 ards, 5 50 ards, and ards. ind the distance H between Main Street and South Main Street. 20 G H Main St Second St South Main St. M THO ppling a atio One alternative approach is to look for ratios in the diagram. ST ead the problem. ecause Main Street, Second Street, and South Main Street are all parallel, the lengths of the segments of the cross streets will be in proportion, so the have the same ratio. ST 2 ppl a ratio. Notice that on ], the distance between South Main Street and Second Street is twice the distance between Second Street and Main Street. So the same will be true for the distances HG and G. HG 5 2 p G Write equation. 5 2 p 20 Substitute Simplif. ST 3 alculate the distance. Line H is perpendicular to both Main Street and South Main Street, so the distance between Main Street and South Main Street is this perpendicular distance, H. H 5 HG G Segment ddition ostulate Substitute Simplif. ST 4 heck page 399 to verif our answer, and confirm that it is the same. 404 hapter 6 Similarit

52 M THO 2 Writing a roportion nother alternative approach is to use a graphic organizer to set up a proportion. ST Make a table to compare the distances. ] ] H Total distance , or 450 artial distance ST 2 Write and solve a proportion. 450 } 50 5 } 20 Write proportion Multipl each side b 2 and simplif. c The distance is 360 ards. TI. MS Use the information on the map. 75 d 225 d 3. WLKING Two people leave points and at the same time. The intend to meet at point at the same time. The person who leaves point walks at a speed of 3 miles per hour. How fast must the person who leaves point walk? 90 d 0.6 mi 0.9 mi a. ind. b. What If? Suppose there is an alle one fourth of the wa from } to } and parallel to }. What is the distance from to the alle along ]? 2. SONING Given the diagram below, eplain wh the three given proportions are true. a } 5 } d a b e a a } a b c 5 d } f a b } a b c 5 d } f d e f b c 4. O NLYSIS student who attempted to solve the problem in ercise 3 claims that ou need to know the length of } to solve the problem. escribe and correct the error that the student made. 5. LG Use the diagram to find the values of and Using lternative Methods 405

53 tension Use after Lesson 6.6 ractals GOL plore the properties of fractals. Ke Vocabular fractal self-similarit iteration HISTOY NOT omputers made it easier to stud mathematical iteration b reducing the time needed to perform calculations. Using fractals, mathematicians have been able to create better models of coastlines, clouds, and other natural objects. fractal is an object that is self-similar. n object is self-similar if one part of the object can be enlarged to look like the whole object. In nature, fractals can be found in ferns and branches of a river. Scientists use fractals to map out clouds in order to predict rain. Man fractals are formed b a repetition of a sequence of the steps called iteration. The first stage of drawing a fractal is considered Stage 0. Helge van Koch ( ) described a fractal known as the Koch snowflake, shown in ample. XML raw a fractal Mandelbrot fractal Use the directions below to draw a Koch snowflake. Starting with an equilateral triangle, at each stage each side is divided into thirds and a new equilateral triangle is formed using the middle third as the triangle side length. Solution STG 0 raw an equilateral triangle with a side length of one unit. STG eplace the middle third of each side with an equilateral triangle. STG 2 epeat Stage with the si smaller equilateral triangles. STG 3 epeat Stage with the eighteen smaller equilateral triangles. 406 hapter 6 Similarit

54 MSUMNT enoit Mandelbrot (b. 924) was the first mathematician to formalize the idea of fractals when he observed methods used to measure the lengths of coastlines. oastlines cannot be measured as straight lines because of the inlets and rocks. Mandelbrot used fractals to model coastlines. XML 2 ind lengths in a fractal Make a table to stud the lengths of the sides of a Koch snowflake at different stages. Stage number dge length Number of edges erimeter } 3 3 p } 9 2 p } p } } 3 92 } } 9 n } 3 n 3 p 4 n 4 n } 3 n 2 at classzone.com TI XMLS and 2 for s. 3. IMT ind the ratio of the edge length of the triangle in Stage 0 of a Koch snowflake to the edge length of the triangle in Stage. How is the perimeter of the triangle in Stage 0 related to the perimeter of the triangle in Stage? plain. 2. MULTI-ST OLM Use the antor set, which is a fractal whose iteration consists of dividing a segment into thirds and erasing the middle third. a. raw Stage 0 through Stage 5 of the antor set. Stage 0 has a length of one unit. b. Make a table showing the stage number, number of segments, segment length, and total length of the antor set. c. What is the total length of the antor set at Stage 0? Stage 20? Stage n? 3. XTN SONS Sierpinski carpet starts with a square with side length one unit. t each stage, divide the square into nine equal squares with the middle square shaded a different color. a. raw Stage 0 through Stage 3 of a Sierpinski arpet. b. plain wh the carpet is said to be self-similar b comparing the upper left hand square to the whole square. c. Make a table to find the total area of the colored squares at Stage 3. tension: ractals 407

55 Investigating g Geometr TIVITY Use before Lesson ilations MTILS graph paper straightedge compass ruler QUSTION How can ou construct a similar figure? XLO onstruct a similar triangle ST ST 2 O O raw a triangle lot the points (, 3), (5, 3), and (5, ) in a coordinate plane. raw n. ST 3 raw ras Using the origin as an endpoint O, draw ] ] O, O, and ] O. ST 4 O O raw equal segments Use a compass to mark a point on ] O so O 5. Mark a point on ] O so O 5. Mark a point on ] O so O 5. raw the image onnect points,, and to form a right triangle. W ONLUSIONS Use our observations to complete these eercises. Measure }, }, }, and }. alculate the ratios } and }. Using this information, show that the two triangles are similar. 2. epeat the steps in the plore to construct nghj so that 3 p O 5 G, 3 p O 5 H, and 3 p O 5 J. 408 hapter 6 Similarit

56 6.7 erform Similarit Transformations efore You performed congruence transformations. Now You will perform dilations. Wh? So ou can solve problems in art, as in. 26. Ke Vocabular dilation center of dilation scale factor of a dilation reduction enlargement transformation, p. 272 dilation is a transformation that stretches or shrinks a figure to create a similar figure. dilation is a tpe of similarit transformation. In a dilation, a figure is enlarged or reduced with respect to a fied point called the center of dilation. The scale factor of a dilation is the ratio of a side length of the image to the corresponding side length of the original figure. In the figure shown, n XYZ is the image of n. The center of dilation is (0, 0) and the scale factor is } XY. O X Y Z KY ONT or Your Notebook oordinate Notation for a ilation You can describe a dilation with respect to the origin with the notation (, ) (k, k), where k is the scale factor. If 0 < k <, the dilation is a reduction. If k >, the dilation is an enlargement. XML raw a dilation with a scale factor greater than IGMS ll of the dilations in this lesson are in the coordinate plane and each center of dilation is the origin. raw a dilation of quadrilateral with vertices (2, ), (4, ), (4, 2), and (, 2). Use a scale factor of 2. Solution irst draw. ind the dilation of each verte b multipling its coordinates b 2. Then draw the dilation. (, ) (2, 2) (2, ) L(4, 2) (4, ) M(8, 2) L 5 M N (4, 2) N(8, 22) (, 2) (2, 22) 6.7 erform Similarit Transformations 409

57 XML 2 Verif that a figure is similar to its dilation triangle has the vertices (4, 24), (8, 2), and (8, 24). The image of n after a dilation with a scale factor of } is n. 2 a. Sketch n and n. b. Verif that n and n are similar. Solution a. The scale factor is less than one, so the dilation is a reduction. (, ) } 2, } 2 2 (4, 24) (2, 22) (8, 2) (4, ) (8, 24) (4, 22) b. ecause and are both right angles, >. Show that the lengths of the sides that include and are proportional. ind the horizontal and vertical lengths from the coordinate plane. } 0 } 4 }2 5 6 } 3 So, the lengths of the sides that include and are proportional. c Therefore, n, n b the SS Similarit Theorem. GUI TI for amples and 2 ind the coordinates of L, M, and N so that n LMN is a dilation of n Q with a scale factor of k. Sketch n Q and n LMN.. (22, 2), Q(2, 0), (0, 2); k (5, 25), Q(0, 25), (0, 5); k XML 3 ind a scale factor HOTO STIKS You are making our own photo stickers. Your photo is 4 inches b 4 inches. The image on the stickers is. inches b. inches. What is the scale factor of the reduction? Solution The scale factor is the ratio of a side length of the sticker image to a side. in. length of the original photo, or}. In simplest form, the scale factor is }. 4 in hapter 6 Similarit

58 ING IGMS Generall, for a center of dilation at the origin, a point of the figure and its image lie on the same ra from the origin. However, if a point of the figure is the origin, its image is also the origin. O X Y Z XML 4 Standardized Test ractice You want to create a quadrilateral GH that is similar to quadrilateral QS. What are the coordinates of H? LIMINT HOIS You can eliminate choice, because ou can tell b looking at the graph that H is in Quadrant I. The point (2, 25) is in Quadrant II. (2, 25) (7, 8) (2, 5) (5, 8) G ΠS Solution etermine if GH is a dilation of QS b checking whether the same scale factor can be used to obtain,, and G from, Q, and. (, ) (k, k) (3, 0) (9, 0) k 5 3 Q(, ) (3, 3) k 5 3 (0, 2) G(0, 6) k 5 3 ecause k is the same in each case, the image is a dilation with a scale factor of 3. So, ou can use the scale factor to find the image H of point S. S(4, 5) H(3 p 4, 3 p 5) 5 H(2, 5) c The correct answer is. HK raw ras from the origin through each point and its image. GUI TI for amples 3 and 4 3. WHT I? In ample 3, what is the scale factor of the reduction if our photo is 5.5 inches b 5.5 inches? 4. Suppose a figure containing the origin is dilated. plain wh the corresponding point in the image of the figure is also the origin. 6.7 erform Similarit Transformations 4

59 6.7 XISS SKILL TI HOMWOK KY 5 WOK-OUT SOLUTIONS on p. WS for s. 5,, and 27 5 STNIZ TST TI s. 2, 3, 2, 22, 28, 30, and 3. VOULY op and complete: In a dilation, the image is? to the original figure. 2. WITING plain how to find the scale factor of a dilation. How do ou know whether a dilation is an enlargement or a reduction? XMLS and 2 on pp for s. 3 8 XML 3 on p. 40 for s. 9 2 WING ILTIONS raw a dilation of the polgon with the given vertices using the given scale factor k. 3. (22, ), (24, ), (22, 4); k (25, 5), (25, 20), (0, 0); k 5 } (, ), (6, ), (6, 3); k (2, 8), (8, 8), (6, 4); k (28, 0), (0, 8), (4, 0), (0, 24); k 5 3 } 8 8. (0, 0), (0, 3), (2, 4), (2, 2); k 5 3 } 2 INTIYING ILTIONS etermine whether the dilation from igure to igure is a reduction or an enlargement. Then find its scale factor XML 4 on p. 4 for MULTIL HOI You want to create a quadrilateral QS that is similar to quadrilateral JKLM. What are the coordinates of S? ΠK 25 L (2, 4) (4, 22) J M (22, 24) (24, 22) O NLYSIS student found the scale factor of the dilation from } to } to be 2 } 5. escribe and } 5 2 } 5 correct the student s error. 42 hapter 6 Similarit

60 5. O NLYSIS student sas that the figure shown represents a dilation. What is wrong with this statement? INTIYING TNSOMTIONS etermine whether the transformation shown is a translation, reflection, rotation, or dilation INING SL TOS ind the scale factor of the dilation of igure to igure. Then give the unknown lengths of igure n 6 8 m p q 3 r 2. MULTIL HOI In the diagram shown, n O is a dilation of no. The length of a median of n O is what percent of the length of the corresponding median of no? 50% 75% 33 } 3 % 200% O 22. SHOT SONS Suppose ou dilate a figure using a scale factor of 2. Then, ou dilate the image using a scale factor of }. escribe the size and 2 shape of this new image. HLLNG escribe the two transformations, the first followed b the second, that combined will transform n into n. 23. (23, 3), (23, ), (0, ) 24. (6, 0), (9, 6), (2, 6) (6, 6), (6, 2), (0, 2) (0, 3), (, 5), (2, 5) 6.7 erform Similarit Transformations 43

61 OLM SOLVING XML 3 on p. 40 for s ILLO VTISMNT billboard advertising agenc requires each advertisement to be drawn so that it fits in a 2-inch b 6-inch rectangle. The agenc uses a scale factor of 24 to enlarge the advertisement to create the billboard. What are the dimensions of a billboard, in feet? 26. OTTY Your potter is used on a poster for a student art show. You want to make postcards using the same image. On the poster, the image is 8 inches in width and 6 inches in height. If the image on the postcard can be 5 inches wide, what scale should ou use for the image on the postcard? 27. SHOWS You and our friend are walking at night. You point a flashlight at our friend, and our friend s shadow is cast on the building behind him. The shadow is an enlargement, and is 5 feet tall. Your friend is 6 feet tall. What is the scale factor of the enlargement? 28. ON-N MTH escribe how ou can use dilations to create the figure shown below. 5 9 at classzone.com 29. MULTI-ST OLM n has vertices (3, 23), (3, 6), and (5, 6). a. raw a dilation of n using a scale factor of 2 } 3. b. ind the ratio of the perimeter of the image to the perimeter of the original figure. How does this ratio compare to the scale factor? c. ind the ratio of the area of the image to the area of the original figure. How does this ratio compare to the scale factor? 30. XTN SONS Look at the coordinate notation for a dilation on page 409. Suppose the definition of dilation allowed k < 0. a. escribe the dilation if 2 < k < 0. b. escribe the dilation if k < 2. c. Use a rotation to describe a dilation with k WOK-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

62 3. SHOT SONS plain how ou can use dilations to make a perspective drawing with the center of dilation as a vanishing point. raw a diagram. O 32. MIOINTS Let } XY be a dilation of } Q with scale factor k. Show that the image of the midpoint of } Q is the midpoint of } XY. 33. SONING In ercise 32, show that } XY i } Q. 34. HLLNG rectangle has vertices (0, 0), (0, 6), (9, 6), and (9, 0). plain how to dilate the rectangle to produce an image whose area is twice the area of the original rectangle. Make a conjecture about how to dilate an polgon to produce an image whose area is n times the area of the original polgon. MIX VIW Simplif the epression. (p. 873) 35. (3 2) 2 ( 2 5) } 2 ab2 (b 2 a)2 37. (a b) 2 2 (a 2 b) 2 ind the distance between each pair of points. (p. 5) 38. (0, 5) and (4, 3) 39. (23, 0) and (2, 4) 40. (22, 24) and (3, 22) VIW repare for Lesson 7. in s ind the value(s) of the variable(s). 4. rea 5 6 in. 2 (p. 49) 42. n > n (p. 256) 43. n Q is isosceles. (p. 303) Y 2 Q ( ) z h in X Z 7 3 in. 708 QUIZ for Lessons ind the value of. (p. 397) raw a dilation of n with the given vertices and scale factor k. (p. 409) 4. (25, 5), (25, 20), (0, 0); k (22, ), (24, ), (22, 4); k XT TI for Lesson 6.7, p. 907 ONLIN QUIZ at classzone.com 45

63 MIX VIW of roblem Solving STT TST TI classzone.com Lessons ON-N The diagram shows the front of a house. What information would ou need in order to show that nwxy, nvxz using the SS Similarit Theorem? 4. SHOT SONS In the diagram, is it alwas, sometimes, or never true that l i l 2 i l 3? plain. 2 2 W X Y l l 2 l 3 V 2. XTN SONS You leave our house to go to the mall. You drive due north 8 miles, due east 7.5 miles, and due north again 2 miles. 8 mi a. plain how to prove that n, n. b. ind. c. ind, the distance between our house and the mall. 3. SHOT SONS The ardon cactus found in the Sonoran esert in Meico is the tallest tpe of cactus in the world. Marco stands 76 feet from the cactus so that his shadow coincides with the cactus shadow. Marco is 6 feet tall and his shadow is 8 feet long. How tall is the ardon cactus? plain. Not drawn to scale 6 ft Mall Your house W 2 mi N S Z 5. GI NSW In the diagram of the roof truss, HK 5 7 meters, KM 5 8 meters, JL meters, and > 2. ind LM to the nearest tenth of a meter. 6. GI NSW You are designing a catalog for a greeting card compan. The catalog features a 2} 4 inch b 5 2 inch photograph of each card. The actual dimensions of a greeting card are 7 inches b 5 inches. What is the scale factor of the reduction? 7. MULTI-ST OLM ectangle has vertices (2, 2), (4, 2), (4, 24), and (2, 24). a. raw rectangle. Then draw a dilation of rectangle using a scale factor of } 5. Label the image QS. 4 b. ind the ratio of the perimeter of the image to the perimeter of the original figure. How does this ratio compare to the scale factor? c. ind the ratio of the area of the image to the area of the original figure. How does this ratio compare to the scale factor? 8 ft 76 ft 46 hapter 6 Similarit

64 6 ig Idea HT SUMMY IG IS Using atios and roportions to Solve Geometr roblems You can use properties of proportions to solve a variet of algebraic and geometric problems. or Your Notebook or eample, in the diagram above, suppose ou know that } 5 }. Then ou can write an of the following relationships. 5 } 5 6 } 8 5 p } } }6 5 5 } } 5 6 } ig Idea 2 Showing that Triangles are Similar You learned three was to prove two triangles are similar. Similarit ostulate SSS Similarit Theorem SS Similarit Theorem If > and >, then n, n. If } 5 } 5 }, then n, n. If > and } 5 }, then n, n. ig Idea 3 Using Indirect Measurement and Similarit You can use triangle similarit theorems to appl indirect measurement in order to find lengths that would be inconvenient or impossible to measure directl. onsider the diagram shown. ecause the two triangles formed b the person and the tree are similar b the Similarit ostulate, ou can write the following proportion to find the height of the tree. height of person }}} length of person s shadow 5 height of tree }} length of tree s shadow You also learned about dilations, a tpe of similarit transformation. In a dilation, a figure is either enlarged or reduced in size. hapter Summar 47

65 6 HT VIW VIW KY VOULY classzone.com Multi-Language Glossar Vocabular practice or a list of postulates and theorems, see pp ratio, p. 356 proportion, p. 358 means, etremes geometric mean, p. 359 scale drawing, p. 365 scale, p. 365 similar polgons, p. 372 scale factor of two similar polgons, p. 373 dilation, p. 409 center of dilation, p. 409 scale factor of a dilation, p. 409 reduction, p. 409 enlargement, p. 409 VOULY XISS op and complete the statement..? is a transformation in which the original figure and its image are similar. 2. If nq, nxyz, then Q } XY 5? } YZ 5? }?. 3. WITING escribe the relationship between a ratio and a proportion. Give an eample of each. VIW XMLS N XISS Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of hapter atios, roportions, and the Geometric Mean pp XML The measures of the angles in n are in the etended ratio of 3:4:5. ind the measures of the angles. Use the etended ratio of 3 : 4 : 5 to label the angle measures as 38, 48, and Triangle Sum Theorem ombine like terms. 5 5 ivide each side b 2. So, the angle measures are 3(58) 5 458, 4(58) 5 608, and 5(58) XMLS, 3, and 6 on pp for s. 4 6 XISS 4. The length of a rectangle is 20 meters and the width is 5 meters. ind the ratio of the width to the length of the rectangle. Then simplif the ratio. 5. The measures of the angles in nuvw are in the etended ratio of ::2. ind the measures of the angles. 6. ind the geometric mean of 8 and hapter 6 Similarit

66 classzone.com hapter eview ractice 6.2 Use roportions to Solve Geometr roblems pp XML In the diagram, } 5 }. ind. 3 } 5 8 } 2 Substitution ropert of qualit Solve for. ross roducts ropert 3 2 XML 2 on p. 365 for s. 7 8 XISS Use the diagram and the given information to find the unknown length. 7. Given N } 5 QM } QL, find. 8. Given } 5 }, find. L 0 M 6 N Use Similar olgons pp XML In the diagram, HG, KLMN. ind the scale factor. rom the diagram, ou can see that }H and } KL correspond. So, the scale factor of HG to KLMN is H } KL 5 2 } } H K G N 24 8 L 2 M XMLS 2 and 4 on pp for s. 9 XISS In ercises 9 and 0, determine whether the polgons are similar. If the are, write a similarit statement and find the scale factor G H X OSTS Two similar posters have a scale factor of 4 : 5. The large poster s perimeter is 85 inches. ind the small poster s perimeter. 20 Y Z 6 8 hapter eview 49

67 6 rove 6.4 HT VIW Triangles Similar b pp XML etermine whether the triangles are similar. If the are, write a similarit statement. plain our reasoning ecause the are right angles, >. the Triangle Sum Theorem, m 5 808, so m and >. Then, two angles of n are congruent to two angles of n. So, n, n. XMLS 2 and 3 on pp for s. 2 4 XISS Use the Similarit ostulate to show that the triangles are similar S U 358 T LL TOW cellular telephone tower casts a shadow that is 72 feet long, while a tree nearb that is 27 feet tall casts a shadow that is 6 feet long. How tall is the tower? 6.5 rove Triangles Similar b SSS and SS pp XML Show that the triangles are similar. Notice that the lengths of two pairs of corresponding sides are proportional. WZ } 5 } VZ } } 5 } 20 5 } 2 YZ 2 3 XZ 30 3 X Y 30 2 Z 4 20 W V The included angles for these sides, XZY and VZW, are vertical angles, so XZY > VZW. Then nxyz, nvwz b the SS Similarit Theorem. XML 4 on p. 39 for s. 5 6 XISS Use the SSS Similarit Theorem or SS Similarit Theorem to show that the triangles are similar T 4.5 U 9 S hapter 6 Similarit

68 classzone.com hapter eview ractice 6.6 Use roportionalit Theorems pp XML etermine whether } M i } LQ. egin b finding and simplifing ratios of lengths determined b } M. NM } 5 } N } } 5 } 24 5 } 2 ML 4 Q 2 M 4 L N ecause NM } ML 5 N } Q, } M is parallel to } LQ b Theorem 6.5, the Triangle roportionalit onverse. XML 2 on p. 398 for s. 7 8 XISS Use the given information to determine whether } i } erform Similarit Transformations pp XML raw a dilation of quadrilateral GHJ with vertices (, ), G(2, 2), H(4, ), and J(2, 2). Use a scale factor of 2. irst draw GHJ. ind the dilation of each verte b multipling its coordinates b 2. Then draw the dilation. G (, ) (2, 2) H (, ) (2, 2) J G(2, 2) (4, 4) H(4, ) (8, 2) J(2, 2) (4, 22) XMLS and 2 on pp for s. 9 2 XISS raw a dilation of the polgon with the given vertices using the given scale factor k. 9. T(0, 8), U(6, 0), V(0, 0); k 5 3 } (6, 0), (3, 9), (0, 0), (3, ); k (8, 2), Q(4, 0), (3, ), S(6, 4); k hapter eview 42

69 6 Solve HT TST the proportion.. 6 } 5 9 } } 5 } b } } } 2a 8 5 } a 2 In ercises 5 7, use the diagram where nq, n. 5. List all pairs of congruent angles. 6. Write the ratios of the corresponding sides in a statement of proportionalit. 7. ind the value of. etermine whether the triangles are similar. If so, write a similarit statement and the postulate or theorem that justifies our answer N 20 M L X 5 8 Y Z K 27 L 6 N M 9 8 J In ercises 3, find the length of } etermine whether the dilation from igure to igure is a reduction or an enlargement. Then find its scale factor SL MOL You are making a scale model of our school s baseball diamond as part of an art project. The distance between two consecutive bases is 90 feet. If ou use a scale factor of } to build our 80 model, what will be the distance around the bases on our model? 422 hapter 6 Similarit

70 6 SOLV QUTI QUTIONS N SIMLIY ILS LG VIW classzone.com lgebra radical epression is simplified when the radicand has no perfect square factor ecept, there is no fraction in the radicand, and there is no radical in a denominator. XML Solve quadratic equations b finding square roots Solve the equation Write original equation. dd 3 to each side ivide each side b 4. 56Ï } 28 Ï } ab 5 Ï } a p Ï } b, so Ï } 28 56Ï } 4 p Ï } Ï } 7 Simplif. XML 2 Simplif quotients with radicals Simplif the epression. a. Î } 0 } 8 b. Î } }5 Solution a. Î } 0 } 8 5 Î } 5 }4 Simplif fraction. 5 Ï} 5 } Ï } 4 Î } a }b 5 Ï} a } Ï } b. 5 Ï} 5 } 2 Simplif. b. Î } }5 5 } Ï } 5 Î } a }b 5 Ï} a } Ï } b. Ï } 5. 5} p Ï} 5 } Multipl numerator and Ï 5 Ï 5 denominator b Ï } 5. 5 Ï} 5 } Multipl fractions. 5 Ï } a p Ï } a 5 a. XISS XML for s. 9 Solve the equation or write no solution ( 2 2 7) (2 2 5) 5 39 XML 2 for s. 0 7 Simplif the epression. 0. Î } 7 }8. Î } 3 }5 2. Î } 24 } Ï } 7 } Ï } 2 4. Î } 75 } Ï } 2 } Ï } } 7. Ï 27 Î } 2 } 42 lgebra eview 423

71 6 Standardized TST TION Scoring ubric ull redit solution is complete and correct artial redit solution is complete but has errors, or solution is without error but is incomplete No redit no solution is given, or solution makes no sense XTN SONS QUSTIONS OLM To find the height of a tree, a student 63 inches in height measures the length of the tree s shadow and the length of his own shadow, as shown. The student casts a shadow 8 inches in length and the tree casts a shadow 477 inches in length. a. plain wh nq, ntqs. b. ind the height of the tree. c. Suppose the sun is a little lower in the sk. an ou still use this method to measure the height of the tree? plain. elow are sample solutions to the problem. ead each solution and the comments in blue to see wh the sample represents full credit, partial credit, or no credit. SML : ull credit solution The reasoning is complete. The proportion and calculations are correct. a. ecause the are both right angles, Q > QTS. lso, Q > Q b the efleive ropert. So, nq, ntqs b the Similarit ostulate. b. } 5 } TS Q TQ 63 } 5 } TS (477) 5 8 p TS 37 5 TS In part (b), the question is answered correctl. In part (c), the reasoning is complete and correct. The height of the tree is 37 inches. c. s long as the sun creates two shadows, I can use this method. ngles and T will alwas be right angles. The measure of Q will change as the sun s position changes, but the angle will still be congruent to itself. So, nq and ntqs will still be similar, and I can write a proportion. 424 hapter 6 Similarit

72 SML 2: artial credit solution In part (a), there is no eplanation of wh the postulate can be applied. In part (b), the proportion is incorrect, which leads to an incorrect solution. In part (c), a partial eplanation is given. a. nq, ntqs b the ngle-ngle Similarit ostulate. b. } 5 } TS Q T 63 } 5 } TS TS The height of the tree is 308 inches. c. s long as the sun creates two shadows, I can use this method because the triangles will alwas be similar. SML 3: No credit solution The reasoning in part (a) is incomplete. In part (b), no work is shown. The answer in part (c) is incorrect. a. The triangles are similar because the lines are parallel and the angles are congruent. b. TS 5 37 inches c. No. The angles in the triangle will change, so ou can t write a proportion. TI ppl the Scoring ubric. student s solution to the problem on the previous page is given below. Score the solution as full credit, partial credit, or no credit. plain our reasoning. If ou choose partial credit or no credit, eplain how ou would change the solution so that it earns a score of full credit. a. Q > TS, and Q is in both triangles. So, nq, ntqs. b. } 5 } QT Q ST 63 } } (477) ø 63.3 The tree is about 63.3 inches tall. c. The method will still work because the triangles will still be similar if the sun changes position. The right angles will sta right angles, and Q is in both triangles, so it does not matter if its measure changes. Standardized Test reparation 425

73 6 Standardized TST TI XTN SONS. Use the diagram. a. plain how ou know that n, n. b. ind the value of n. c. The perimeter of n is 22. What is the perimeter of n? Justif our answer n 2. On the easel shown at the right, } i } H i } G, and }G > }. a. ind,, and. Justif our answer. b. On the easel, } M is a support bar attached to }, }H, and } G. On this support bar, N 5 0 inches. ind the length of } M to the nearest inch. Justif our answer. c. The support bar } M bisects }, } H, and } G. oes this mean that polgons MNH and MG are similar? plain. in. G 30 in. H M N 3. handmade rectangular rug is available in two sizes at a rug store. small rug is 24 inches long and 6 inches wide. large rug is 36 inches long and 24 inches wide. a. re the rugs similar? If so, what is the ratio of their corresponding sides? plain. b. ind the perimeter and area of each rug. Then find the ratio of the perimeters (large rug to small rug) and the ratio of the areas (large rug to small rug). c. It takes 250 feet of wool arn to make square foot of either rug. How man inches of arn are used for each rug? plain. d. The price of a large rug is.5 times the price of a small rug. The store owner wants to change the prices for the rugs, so that the price for each rug is based on the amount of arn used to make the rug. If the owner changes the prices, about how man times as much will the price of a large rug be than the price of a small rug? plain. 4. In the diagram shown at the right, ] OQ passes through the origin. a. plain how ou know that nos, noq. b. ind the coordinates of point Q. Justif our answer. c. The -coordinate of a point on ] OQ is a. Write the -coordinate of this point in terms of a. Justif our answer. (5, 3) O S(5, 0) Π(9, 0) 426 hapter 6 Similarit

74 STT TST TI classzone.com MULTIL HOI 5. If nq, nstu, which proportion is not necessaril true? } Q 5 } ST } Q 5 } Q TU SU TU } 5 } Q } Q 5 } ST SU TU SU GI NSW 8. ind the value of. J K 4 L 9 N M 6. On a map, the distance between two cities is 2} 3 inches. The scale on the map 4 is in.:80 mi. What is the actual distance between the two cities? 60 mi 80 mi 200 mi 220 mi 7. In the diagram, what is the scale factor of the dilation from nq to ntuv? 9. In the diagram below, nqm, nnm, and }M > } Q. If N 5 2, find M. N M 0. Given G 5 0, find H. Π3 H 2 U G 20 T 2 V } } 2 3. In an acute isosceles triangle, the measures of two of the angles are in the ratio 4 :. ind the measure of a base angle in the triangle. 2 3 SHOT SONS 2. On a school campus, the gm is 400 feet from the art studio. a. Suppose ou draw a map of the school campus using a scale of } inch: 00 feet. How far will the gm be from the art studio 4 on our map? b. Suppose ou draw a map of the school campus using a scale of } inch : 00 feet. Will the distance from the gm to the art studio on 2 this map be greater than or less than the distance on the map in part (a)? plain. 3. ectangles and GH are similar, and the ratio of to is :3. In each rectangle, the length is twice the width. The area of is 32 square inches. ind the length, width, and area of GH. plain. Standardized Test ractice 427

75 UMULTIV VIW hapters 6 ind m 2 if and 2 are (a) complementar angles and (b) supplementar angles. (p. 24). m m m m Solve the equation and write a reason for each step. (p. 05) ( 2 3) ( 2) 5 25 State the postulate or theorem that justifies the statement. (pp. 47, 54) 8. > > 6 0. m 3 m > > 3 3. m 7 m p 2 l m The variable epressions represent the angle measures of a triangle. ind the measure of each angle. Then classif the triangle b its angles. (p. 27) 4. m m m 5 (3 2 5)8 m 5 38 m 5 28 m 5 ( 5)8 m 5 48 m 5 ( 2 5)8 m 5 ( 2 20)8 etermine whether the triangles are congruent. If so, write a congruence statement and state the postulate or theorem ou used. (pp. 234, 240, 249) V 9. Y W X S Z M N ind the value of. (pp. 295, 303, 30) Y X W 2 9 Z ( 2)8 (3 2 28)8 etermine whether the triangles are similar. If the are, write a similarit statement and state the postulate or theorem ou used. (pp. 38, 388) G Y 6 Z H 33 H 368 V W X N G S umulative eview: hapters 6

76 26. OITS compan s profits for two ears are shown in the table. lot and connect the points (, ). Use the Midpoint ormula to estimate the compan s profits in (ssume that profits followed a linear pattern.) (p. 5) Years since 2000, 5 rofit, (in dollars) 2,000 36, TNNIS MMSHI The graph at the right models the accumulated cost for an individual adult tennis club membership for several months. (p. 80) a. Write an equation of the line. b. Tell what the slope and -intercept mean in this situation. c. ind the accumulated cost for one ear. ost (dollars) 750 (9, 78) (0, 250) Month OO Write a two-column proof or a paragraph proof. (pp. 234, 240, 249) 28. GIVN c } G > } HJ, } MH > } KG, 29. GIVN c } i } }M } J, } KJ } J } > } OV c nhm > njgk OV c n > n G H J L M K 30. OMMUNITY NT building committee needs to choose a site for a new communit center. The committee decides that the new center should be located so that it is the same distance from each of the three local schools. Use the diagram to make a sketch of the triangle formed b the three schools. plain how ou can use this triangle to locate the site for the new communit center. (p. 303) ranklin Washington Monroe 3. GOGHY The map shows the distances between three cities in North akota. escribe the range of possible distances from owman to llendale. (p. 328) 32. LN You send 2 photos to a compan that makes personalized wall calendars. The compan enlarges the photos and inserts one for each month on the calendar. ach photo is 4 inches b 6 inches. The image for each photo on the calendar is 0 inches b 5 inches. What is the scale factor of the enlargement? (p. 409) umulative eview: hapters 6 429