Similar Right Triangles
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1 9.3 EX EENIL KNOWLEGE N KILL G.8. G.8. imilar igt riangles Essential Question How are altitudes and geometric means of rigt triangles related? Writing a onjecture Work wit a partner. a. Use dnamic geometr software to construct rigt, as sown. raw so tat it is an altitude from te rigt angle to te potenuse of Points (0, ) (8, 0) (0, 0) (., 3.6) egments = 9.3 = 8 = MKING MHEMIL GUMEN o be proficient in mat, ou need to understand and use stated assumptions, definitions, and previousl establised results in constructing arguments. b. e geometric mean of two positive numbers a and b is te positive number tat satisfies a =. is te geometric mean of a and b. b Write a proportion involving te side lengts of and so tat is te geometric mean of two of te oter side lengts. Use similar triangles to justif our steps. c. Use te proportion ou wrote in part (b) to find. d. Generalize te proportion ou wrote in part (b). en write a conjecture about ow te geometric mean is related to te altitude from te rigt angle to te potenuse of a rigt triangle. Work wit a partner. Use a spreadseet to find te aritmetic mean and te geometric mean of several pairs of positive numbers. ompare te two means. Wat do ou notice? ommunicate Your nswer omparing Geometric and ritmetic Means a b ritmetic Mean Geometric Mean How are altitudes and geometric means of rigt triangles related? ection 9.3 imilar igt riangles 81
2 9.3 Lesson Wat You Will Learn ore Vocabular geometric mean, p. 8 Previous altitude of a triangle similar figures Identif similar triangles. olve real-life problems involving similar triangles. Use geometric means. Identifing imilar riangles Wen te altitude is drawn to te potenuse of a rigt triangle, te two smaller triangles are similar to te original triangle and to eac oter. eorem eorem 9.6 igt riangle imilarit eorem If te altitude is drawn to te potenuse of a rigt triangle, ten te two triangles formed are similar to te original triangle and to eac oter.,, and. Proof E., p. 88 Identifing imilar riangles Identif te similar triangles in te diagram. U ketc te tree similar rigt triangles so tat te corresponding angles and sides ave te same orientation. U U U U Monitoring Progress Help in Englis and panis at igideasmat.com Identif te similar triangles. 1. Q. E H F G 8 apter 9 igt riangles and rigonometr
3 olving eal-life Problems Modeling wit Matematics roof as a cross section tat is a rigt triangle. e diagram sows te approimate dimensions of tis cross section. Find te eigt of te roof. Y. m 3.1 m Z 6.3 m W X 1. Understand te Problem You are given te side lengts of a rigt triangle. You need to find te eigt of te roof, wic is te altitude drawn to te potenuse.. Make a Plan Identif an similar triangles. en use te similar triangles to write a proportion involving te eigt and solve for. 3. olve te Problem Identif te similar triangles and sketc tem. Z Z OMMON EO Notice tat if ou tried to write a proportion using XYW and YZW, ten tere would be two unknowns, so ou would not be able to solve for. 3.1 m X Y W. m Y W X 6.3 m XYW YZW XZY ecause XYW XZY, ou can write a proportion. 3.1 m Y. m YW ZY = XY XZ orresponding side lengts of similar triangles are proportional.. = ubstitute..7 Multipl eac side b.. e eigt of te roof is about.7 meters.. Look ack ecause te eigt of te roof is a leg of rigt YZW and rigt XYW, it sould be sorter tan eac of teir potenuses. e lengts of te two potenuses are YZ =. and XY = 3.1. ecause.7 < 3.1, te answer seems reasonable. Monitoring Progress Find te value of. Help in Englis and panis at igideasmat.com 3. E 3 G H F. J 13 1 K L M ection 9.3 imilar igt riangles 83
4 Using a Geometric Mean ore oncept Geometric Mean e geometric mean of two positive numbers a and b is te positive number tat satisfies a = b. o, = ab and = ab. Finding a Geometric Mean Find te geometric mean of and 8. = ab efinition of geometric mean = 8 ubstitute for a and 8 for b. = 8 ake te positive square root of eac side. = Factor. = implif. e geometric mean of and 8 is In rigt, altitude is drawn to te potenuse, forming two smaller rigt triangles tat are similar to. From te igt riangle imilarit eorem, ou know tat. ecause te triangles are similar, ou can write and simplif te following proportions involving geometric means. = = = = = = eorems eorem 9.7 Geometric Mean (ltitude) eorem In a rigt triangle, te altitude from te rigt angle to te potenuse divides te potenuse into two segments. e lengt of te altitude is te geometric mean of te lengts of te two segments of te potenuse. Proof E. 1, p. 88 = eorem 9.8 Geometric Mean (Leg) eorem In a rigt triangle, te altitude from te rigt angle to te potenuse divides te potenuse into two segments. e lengt of eac leg of te rigt triangle is te geometric mean of te lengts of te potenuse and te segment of te potenuse tat is adjacent to te leg. = = Proof E., p apter 9 igt riangles and rigonometr
5 Using a Geometric Mean OMMON EO In Eample (b), te Geometric Mean (Leg) eorem gives = ( + ), not = ( + ), because te side wit lengt is adjacent to te segment wit lengt. Find te value of eac variable. a. 6 3 b. a. ppl te Geometric Mean b. ppl te Geometric Mean (ltitude) eorem. (Leg) eorem. = 6 3 = ( + ) = 18 = 7 = 18 = 1 = 9 = 1 = 3 e value of is 1. e value of is 3. Using Indirect Measurement o find te cost of installing a rock wall in our scool gmnasium, ou need to find te eigt of te gm wall. You use a cardboard square to line up te top and bottom om of te gm wall. Your friend measures te vertical distance from te ground to our ee and te orizontal distance from ou to te gm wall. pproimate te eigt of te gm wall. te Geometric Mean (ltitude) eorem, ou know tat 8. is te geometric mean of w and. 8. = w Geometric Mean (ltitude) eorem 7. = w quare = w ivide eac side b. e eigt of te wall is + w = + 1. = 19. feet. 8. ft w ft ft 9 Monitoring Progress Find te geometric mean of te two numbers. Help in Englis and panis at igideasmat.com. 1 and and and Find te value of in te triangle at te left. 9. WH IF? In Eample, te vertical distance from te ground to our ee is. feet and te distance from ou to te gm wall is 9 feet. pproimate te eigt of te gm wall. ection 9.3 imilar igt riangles 8
6 9.3 Eercises namic olutions available at igideasmat.com Vocabular and ore oncept eck 1. OMPLEE HE ENENE If te altitude is drawn to te potenuse of a rigt triangle, ten te two triangles formed are similar to te original triangle and.. WIING In our own words, eplain geometric mean. Monitoring Progress and Modeling wit Matematics In Eercises 3 and, identif te similar triangles. (ee Eample 1.) 3. F E In Eercises 11 18, find te geometric mean of te two numbers. (ee Eample 3.) and and 16 H G and 0 1. and and and 8. M and and L N K In Eercises 10, find te value of. (ee Eample.). 6. Q W Y X Z In Eercises 19 6, find te value of te variable. (ee Eample.) E H F G b ft 1.8 ft.8 ft.6 ft 3. ft. z ft 86 apter 9 igt riangles and rigonometr
7 EO NLYI In Eercises 7 and 8, describe and correct te error in writing an equation for te given diagram. 7. z MHEMIL ONNEION In Eercises 31 3, find te value(s) of te variable(s). 31. a b w v z = w (w + v) e g f d z EONING Use te diagram. ecide wic proportions are true. elect all tat appl. z 3 d = f MOELING WIH MHEMI In Eercises 9 and 30, use te diagram. (ee Eample.) = = = = 7. ft. ft 6 ft 9. ft E. 9 E You want to determine te eigt of a monument at a local park. You use a cardboard square to line up te top and bottom of te monument, as sown at te above left. Your friend measures te vertical distance from te ground to our ee and te orizontal distance from ou to te monument. pproimate te eigt of te monument. 30. Your classmate is standing on te oter side of te monument. e as a piece of rope staked at te base of te monument. e etends te rope to te cardboard square se is olding lined up to te top and bottom of te monument. Use te information in te diagram above to approimate te eigt of te monument. o ou get te same answer as in Eercise 9? Eplain our reasoning. 36. NLYZING ELIONHIP You are designing a diamond-saped kite. You know tat =.8 centimeters, = 7 centimeters, and = 8.8 centimeters. You want to use a straigt crossbar. bout ow long sould it be? Eplain our reasoning. 37. NLYZING ELIONHIP Use te Geometric Mean eorems (eorems 9.7 and 9.8) to find and. 0 1 ection 9.3 imilar igt riangles 87
8 38. HOW O YOU EE I? In wic of te following triangles does te Geometric Mean (ltitude) eorem (eorem 9.7) appl? 0. MKING N GUMEN Your friend claims te geometric mean of and 9 is 6, and ten labels te triangle, as sown. Is our friend correct? Eplain 9 our reasoning. 6 In Eercises 1 and, use te given statements to prove te teorem. Given is a rigt triangle. ltitude is drawn to potenuse. 1. POVING HEOEM Prove te Geometric Mean (ltitude) eorem (eorem 9.7) b sowing tat =. 39. POVING HEOEM Use te diagram of. op and complete te proof of te Ptagorean eorem (eorem 9.1). Given In, is a rigt angle. Prove c = a + b EMEN 1. In, is a rigt angle.. raw a perpendicular segment (altitude) from to. EON 1.. Perpendicular Postulate (Postulate 3.) 3. ce = a and cf = b 3.. ce + b = + b. ddition Propert of Equalit. ce + cf = a + b. 6. c(e + f ) = a + b e + f = 7. egment ddition Postulate (Postulate 1.) 8. c c = a + b c = a + b 9. implif. b f a c e. POVING HEOEM Prove te Geometric Mean (Leg) eorem (eorem 9.8) b sowing tat = and =. 3. IIL HINKING raw a rigt isosceles triangle and label te two leg lengts. en draw te altitude to te potenuse and label its lengt. Now, use te igt riangle imilarit eorem (eorem 9.6) to draw te tree similar triangles from te image and label an side lengt tat is equal to eiter or. Wat can ou conclude about te relationsip between te two smaller triangles? Eplain our reasoning.. HOUGH POVOKING e aritmetic mean and geometric mean of two nonnegative numbers and are sown. aritmetic mean = + geometric mean = Write an inequalit tat relates tese two means. Justif our answer.. POVING HEOEM Prove te igt riangle imilarit eorem (eorem 9.6) b proving tree similarit statements. Given is a rigt triangle. ltitude is drawn to potenuse. Prove,, Maintaining Matematical Proficienc olve te equation for. (kills eview Handbook) = 7. 9 = 8. 9 = 78 eviewing wat ou learned in previous grades and lessons = apter 9 igt riangles and rigonometr
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