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ewer/Lesson.htmCHAPTER11SimilarityOBJECTIVESIn this chapter you willNobody can draw a line that isnot a boundary line, every lineseparates a unity into a multiplicity. Inaddition, every closed contour no matterwhat its shape, pure circle or whimsical splashaccidental in form, evokes the sensation of “inside”and “outside,” followed quickly by the suggestion of“nearby” and “far off,” of object and background.M. C. ESCHERPath of Life I, M. C. Escher, 1958 2002 Cordon Art B.V.–Baarn–Holland.All rights reserved. 2008 Key Curriculum Pressreview ratio andproportiondefine similar polygonsand solidsdiscover shortcuts forsimilar triangleslearn about area andvolume relationships insimilar polygons and solidsuse the definition ofsimilarity to solveproblems

ewer/Lesson.htmProportion and ReasoningWorking with similar geometric figures involves ratios and proportions. A ratio isan expression that compares two quantities by division. You can write the ratio ofquantity a to quantity b in these three ways:a to ba:bThis book will write ratios in fraction form. As with fractions, you can multiplyor divide both parts of a ratio by the same number to get an equivalent ratio.A proportion is a statement of equality between two ratios. The equalityis an example of a proportion. Proportions are useful for solving problemsinvolving comparisons.EXAMPLE AIn a photograph, Dan is 2.5 inches tall and his sister Emma is 1.5 inches tall.Dan’s actual height is 70 inches. What is Emma’s actual height?The ratio of Dan’s height to Emma’s height is the same in real life as it is in thephoto. Let x represent Emma’s height. Set up a proportion.SolutionDan’s height inphotographDan’s actualheightEmma’s heightin photographEmma’s actualheightFind Emma’s height by solving for x.Original proportion.x 702.5x 105x 42Multiply both sides by x.Multiply both sides by 1.5.Divide both sides by 2.5.Emma is 42 inches tall.You could also have solved this proportion by inverting both ratios so that x isin the numerator, and then multiplying both sides by 70. Inverting both ratios isthe same as comparing Emma’s height to Dan’s instead of Dan’s to Emma’s.There are still other ways to use proportional reasoning to solve Example A. Forinstance, Dan is actually, or 28 times as tall as he is in the photo, so Emma’sheight must be 28 · 1.5, or 42 inches. You could also have solved the proportion, because the ratio of Dan’s actual height to his height in the photo is equalto the ratio of Emma’s actual height to her height in the photo.578CHAPTER 11Similarity 2008 Key Curriculum Press

ewer/Lesson.htmSome proportions require more algebra to solve.EXAMPLE BSolutionSolve The least common denominator is 10(x 40).10(x 40) · 10(x 40) ·Multiply both sides by the leastcommon denominator.10(3x13) 23(x 40)Simplify each side by dividing to “clear”the fractions.30x130 23x 920Distribute on the left side and on theright side of the equation.7x 1050Subtract 23x from both sides and add130 to both sides.x 150Divide both sides by 7.EXERCISES1. Look at the rectangle at right. Find the ratio of the shaded area tothe area of the whole figure. Find the ratio of the shaded area tothe unshaded area.2. Use the figure below to find these ratios:and.3. Consider these right triangles.a. Find the ratio of the perimeter of RSH to the perimeter ofb. Find the ratio of the area of RSH to the area of MFL.MFL.In Exercises 4–12, solve the proportion.4.5.6.7.8.9.10.11.12. 2008 Key Curriculum PressUSING YOUR ALGEBRA SKILLS 11 Proportion and Reasoning579

ewer/Lesson.htmIn Exercises 13–16, use a proportion to solve the problem.13. Application A car travels 106 miles on 4 gallons of gas. How far can it go on a fulltank of 12 gallons?14. Application Ernie is a baseball pitcher. He gave up 34 runs in 152 innings lastseason. What is Ernie’s earned run average—the number of runs he would give upin 9 innings? Give your answer to the nearest hundredth of a unit.15. Application The floor plan of a house is drawn tothe scale of in. 1 ft. The master bedroommeasures 3 in. byin. on the blueprints. What isthe actual size of the room?16. Altor and Zenor are ambassadors from Titan, thelargest moon of Saturn. The sum of the lengths ofany Titan’s antennae is a direct measure of thatTitan’s age. Altor has antennae with lengths 8 cm,10 cm, 13 cm, 16 cm, 14 cm, and 12 cm. Zenor is130 years old, and her seven antennae have anaverage length of 17 cm. How old is Altor?, which of these statements are also true? Assume that a, b, c, and d are17. Ifall nonzero values. For each proportion, use algebra to show it is true or give acounterexample to prove it is false.a. ad bcb. ac bdc.d.e.f.18. The sequence 6, 15, 24, 33, 42, . . . is an example of an arithmetic sequence—eachterm is generated by adding a constant, in this case 9, to the previous term.The sequence 6, 12, 24, 48, 96, . . . is an example of a geometric sequence—eachterm is generated by multiplying the previous term by a constant, in this case 2.Find the missing terms assuming each pattern is an arithmetic sequence, and thenfind the missing terms assuming each pattern is a geometric sequence.a. 10, , 40, , . . .b. 2, , 50, , . . .c. 4, , 36, , . . .19. Consider the pattern 10, x, 40, . . .a. If the pattern is an arithmetic sequence, the value of x is called thearithmetic mean of 10 and 40. Use your algebra skills to explain howto calculate the arithmetic mean of any two numbers.b. If the pattern is a geometric sequence, the positive value of x is called thegeometric mean of 10 and 40. Use your algebra skills to explain how tocalculate the geometric mean of any two positive numbers.c. For any two positive numbers, the ratio of the smaller number to their geometricmean is equal to the ratio of the geometric mean to the larger number. Find theformula for the geometric mean of a and b by solving the proportionfor c.Are there any values for which this formula isn’t true?d. Use this formula to find the geometric mean of 2 and 50, and of 4 and 36.580CHAPTER 11Similarity 2008 Key Curriculum Press

ewer/Lesson.htmL E S S O N11.2Similar TrianglesIn Lesson 11.1, you concluded that you must know about both the angles and thesides of two quadrilaterals in order to make a valid conclusion about their similarity.Life is change.Growth isoptional.Choose wisely.KAREN KAISER CLARKHowever, triangles are unique. Recall from Chapter 4 that you found four shortcutsfor triangle congruence: SSS, SAS, ASA, and SAA. Are there shortcuts for trianglesimilarity as well? Let’s first look for shortcuts using only angles.The figures below illustrate that you cannot conclude that two triangles are similargiven that only one set of corresponding angles are congruent.AD, butABC is not similar to DEF or to DFE.How about two sets of congruent angles?Is AA a Similarity Shortcut?If two angles of one triangle are congruent to two angles of another triangle, mustthe two triangles be similar?a compassa rulerStep 1Draw any triangle ABC.Step 2Construct a second triangle, DEF, withtrue about C and F ? Why?Step 3Carefully measure the lengths of the sides of both triangles. Compare the ratiosof the corresponding sides. IsStep 4Compare your results with the results of others near you. You should be ready tostate a conjecture.DA andEB. What will beAA Similarity ConjectureIfthenangles of one triangle are congruent toangles of another triangle,As you may have guessed from Step 2 of the investigation, there is no need toinvestigate the AAA, ASA, or SAA Similarity Conjectures. Thanks to the TriangleSum Conjecture, or more specifically the Third Angle Conjecture, the AA SimilarityConjecture is all you need. 2008 Key Curriculum PressLESSON 11.2 Similar Triangles589

ewer/Lesson.htmNow let’s look for shortcuts for similarity that use only sides. The figures belowillustrate that you cannot conclude that two triangles are similar given that two setsof corresponding sides are proportional.but GWB is not similar to JFK.How about all three sets of corresponding sides?Is SSS a Similarity Shortcut?a compassa straightedgea protractorIf three sides of one triangle are proportional to the three sides of another triangle,must the two triangles be similar?Draw any triangle ABC. Then construct a second triangle, DEF, whose side lengthsare a multiple of the original triangle. (Your second triangle can be larger orsmaller.)Compare the corresponding angles of the two triangles. Compare your results withthe results of others near you and state a conjecture.SSS Similarity ConjectureIf the three sides of one triangle are proportional to the three sides of anothertriangle, then the two triangles areCareerSimilarity plays an important part in thedesign of cars, trucks, and airplanes,which is done with small-scale drawingsand models.This model airplane is about to be testedin a wind tunnel.590CHAPTER 11Similarity 2008 Key Curriculum Press

ewer/Lesson.htmSo SSS, AAA, ASA, and SAA are shortcuts for triangle similarity. That leaves SASand SSA as possible shortcuts to consider.Is SAS a Similarity Shortcut?a compassa protractora rulerIs SAS a shortcut for similarity? Try to construct two different triangles that are notsimilar but have two pairs of sides proportional and the pair of included anglesequal in measure.Compare the measures of corresponding sides and corresponding angles. Shareyour results with others near you and state a conjecture.SAS Similarity ConjectureIf two sides of one triangle are proportional to two sides of another triangleand, then theOne question remains: Is SSA a shortcut for similarity? Recall from Chapter 4 thatSSA did not work for congruence because you could create two different triangles.Those two different triangles were neither congruent nor similar. So, no, SSA is nota shortcut for similarity.keymath.com/DGFor an interactive version of all the investigations in this lesson, see theDynamic Geometry Exploration Similar Triangles at www.keymath.com/DG .EXERCISESFor Exercises 1–14, use your new conjectures. All measurements are in centimeters.1. g 2008 Key Curriculum Press2. h ,k 3. m LESSON 11.2 Similar Triangles591

ewer/Lesson.htm4. n s ,7. Is PHYYHT?Is PTY a right triangle?Explain why or why not.10. ORf UE NT,g 13. FROG is a trapezoid.FRG?Is RGOIs GOFRFO?RFS?Why is GOSt ,s 592CHAPTER 11Similarity5. Is AULMST?Explain why or why not.6. Is MOYNOT?Explain why or why not.8. Why is TMRMHR?Find x, y,and h.9. TA URIs QTATUR?ARU?Is QATWhy is QTAQUR?e 11. Is THUIs HTUp ,q THMGDU?DGU?14. TOAD is a trapezoid.w ,x 12. Why is SUNr ,s TAN?15. Find x and y. 2008 Key Curriculum Press

ewer/Lesson.htmReview16. In the figure below right, find the radius, r, of one of the small circles in terms ofthe radius, R, of the large circle.Rr17. Application Phoung volunteers at anSPCA that always houses 8 dogs. Shenotices that she uses seven 35-pound bagsof dry dog food every two months. A new,larger SPCA facility that houses 20 dogswill open soon. Help Phoung estimate theamount of dry dog food that the facilityshould order every three months. Explainyour reasoning.18. Application Ramon and Sabina areoceanography students studying thehabitat of a Hawaiian fish calledHumuhumunukunukuapua‘a. They aregoing to use the capture-recapture methodto determine the fish population. Theyfirst capture and tag 84 fish, which theyrelease back into the ocean. After oneweek, Ramon and Sabina catch another 64.Only 12 have tags. Can you estimate thepopulation of Humuhumunukunukuapua‘a? 2008 Key Curriculum PressLESSON 11.2 Similar Triangles593