12.3 Using Proportional Relationships

Transcription

1 Name lass Date 12.3 Using Proportional Relationsips ssential Question: How can you use similar triangles to solve problems? Resource Locker xplore xploring Inirect Measurement In tis xplore, you will consier ow to fin eigts, lengts, or istances tat are too great to be measure irectly, tat is, wit measuring tools like rulers. Inirect measurement involves using te properties of similar triangles to measure suc eigts or istances. During te ay sunligt creates saows, as sown in te figure below. Te ase segment represents te ray of sunligt. Wat kin of triangle is forme by te flagpole, its saow, an te ray of sunligt? Hougton Mifflin Harcourt Publising ompany Image reits: naptalina/istockpoto.com Suppose te sun is sining, an you are staning near a flagpole, but out of its saow. ou will cast a saow as well. ou can assume tat te rays of te sun are parallel. Wat o you know about te two triangles forme? xplain your reasoning. In te iagram, wat eigts or lengts o you alreay know? D Wat eigts or lengts can be measure irectly? Moule Lesson 3

2 Reflect 1. How coul you use similar triangles to measure te eigt of te flagpole inirectly? xplain 1 Fining an Unknown Heigt xample 1 Fin te inicate imension using te measurements sown in te figure an te properties of similar triangles. In orer to fin te eigt of a palm tree, you measure te tree s saow an, at te same time of ay, you measure te saow cast by a meter stick tat you ol at a rigt angle to te groun. Fin te eigt of te tree. 1 m ecause ǁ,. ll rigt angles are congruent, so. So. 1.6 m 7.2 m Set up proportion. _ = _ Substitute. _ 7.2 = 1_ 1.6 Multiply eac sie by 7.2. = 7.2 ( 1_ 1.6) Simplify. = 4.5 Te tree is 4.5 meters ig. Si is 72 inces tall. To measure a flagpole, Si stans near te flag. Si s frien Mirana measures te lengts of Si s saow an te flagpole s saow. Fin te eigt of te flagpole. Te triangles are similar by te Similarity riterion. Set up proportion. Substitute. Multiply eac sie by in. 48 in. 128 in. eigt = saow person s eigt person s saow _ 72 = _ 48 = 72 ( _ 48 ) Hougton Mifflin Harcourt Publising ompany Simplify. x = Te flagpole is tall. Moule Lesson 3

3 Reflect 2. In te tree example, ow can you ceck tat your answer is reasonable? our Turn 3. Liam is 6 feet tall. To fin te eigt of a tree, e measures is saow an te tree s saow. Te measurements of te two saows are sown. Fin te eigt of te tree. Liam Tree 6 ft 8 ft 28 ft xplain 2 Fining an Unknown Distance In real-worl situations, you may not be able to measure an object irectly because tere is a pysical barrier separating you from te object. ou can use similar triangles in tese situations as well. xample 2 xplain ow to use te information in te figure to fin te inicate istance. iker wants to fin te istance across a canyon. Se locates points as escribe. Hougton Mifflin Harcourt Publising ompany 1. Se ientifies a lanmark at. Se places a marker () irectly across te canyon from. 2. t, se turns 90 away from an walks 400 feet in a straigt line. Se places a marker () at tis location. 3. Se continues walking anoter 600 feet, an places a marker (W) at tis location. 4. Se turns 90 away from te canyon an walks until te marker aligns wit. Se places a marker (V) at tis location an measures _ WV. VW (ll rigt angles are congruent) an VW (Vertical angles are congruent). So, VW ~ by te Similarity riterion. _ VW = _ W, So _ 327 = _ , or _ 327 = 2_ 3 Ten = 327 ( 2_ 3) = 218. Te istance across te canyon is 218 feet. 327 ft W V 600 ft 400 ft Moule Lesson 3

4 To fin te istance across te gorge, a stuent ientifies points as sown in te figure. Fin. JKL ~ NML by te Triangle Similarity Teorem. JK_ NM = _ KL J K 24 m L 42 m M 35 m _ 35 = _ 24 N = _ 42 = _ 7 = _ 7 = Te istance across te gorge is. Reflect 4. In te example, wy is JLK NLM? Hour Turn 5. To fin te istance across a stream, Levi locate points as sown in te figure. Use te given information to fin. laborate 6. Discussion Suppose you want to elp a frien prepare for solving inirect measurement problems. Wat topics woul you suggest tat your frien review? 12 m 6 m 12 m D Hougton Mifflin Harcourt Publising ompany 7. ssential Question eck-in ou are given a figure incluing triangles tat represent a real-worl situation. Wat is te first step you soul take to fin an unknown measurement? Moule Lesson 3

5 valuate: Homework an Practice 1. Fining istances using similar triangles is calle. Use similar triangles an to fin te missing eigt Online Homework Hints an Help xtra Practice 6 ft 60 ft 15 ft 156 ft 5.5 ft 16.5 ft ft 14 ft 4 ft 208 ft 15.2 ft 3.8 ft Use similar triangles FG an IHG to fin te missing istance Hougton Mifflin Harcourt Publising ompany F 48 m G 60 m H 80 m I 45 m F 78 m G H 180 m I 18 m F H 27 m G m I 9. F m m G 64.8 m H I Moule Lesson 3

6 10. To fin te eigt of a inosaur in a museum, mir place a mirror on te groun 40 feet from its base. Ten e steppe back 4 feet so tat e coul see te top of te inosaur in te mirror. mir s eyes were approximately 5 feet 6 inces above te groun. Wat is te eigt of te inosaur? 5 ft 6 in. 4 ft 40 ft 11. Jenny is 5 feet 2 inces tall. To fin te eigt of a ligt pole, se measure er saow an te pole s saow. Wat is te eigt of te pole? 5 ft 2 in ft 7 ft 9 in. 12. stuent wante to fin te eigt of a statue of a pineapple in Nambour, ustralia. Se measure te pineapple s saow an er own saow. Te stuent s eigt is 5 feet 4 inces. Wat is te eigt of te pineapple? D 13. To fin te eigt of a flagpole, asey measure er own saow an te flagpole s saow. Given tat asey s eigt is 5 feet 4 inces, wat is te eigt of te flagpole? 1 2 ft 2 8 ft 9 in. F Hougton Mifflin Harcourt Publising ompany 5 ft 4 in. 3 ft 14 ft 3 in. Moule Lesson 3

7 city is planning an outoor concert for an Inepenence Day celebration. To ol speakers an ligts, a crew of tecnicians sets up a scaffol wit two platforms by te stage. Te first platform is 8 feet 2 inces off te groun. Te secon platform is 7 feet 6 inces above te first platform. Te saow of te first platform stretces 6 feet 3 inces across te groun. 7 ft 6 in. 8 ft 2 in. 6 ft 3 in. D 14. xplain wy is similar to D. (Hint: rays of ligt are parallel.) 15. Fin te lengt of te saow of te secon platform in feet an inces to te nearest inc. Hougton Mifflin Harcourt Publising ompany 16. tecnician is 5 feet 8 inces tall. Te tecnician is staning on top of te secon platform. Fin te lengt s of te saow tat is cast by te scaffol an te tecnician to te nearest inc. Moule Lesson 3

8 17. To fin te istance across a lake, you locate points as sown in te figure. xplain ow to use tis information to fin. 500 ft U 300 ft 400 ft V 800 ft 600 ft 18. In orer to fin te eigt of a cliff, you stan at te bottom of te cliff, walk 60 feet from te base, an place a mirror on te groun. Ten you face te cliff an step back 5 feet so tat can see te top of te cliff in te mirror. ssuming your eyes are 6 feet above te groun, explain ow to use tis information to fin te eigt of te cliff. (Te angles marke congruent are congruent because of te nature of te reflection of ligt in a mirror.) P J 6 ft M Q 5 ft 60 ft K Mirror 19. To fin te eigt of a tree, rian measures te tree s saow an ten is saow. Wic proportion coul rian use to fin te eigt of te tree? Select all tat apply.. _ DF = _ F. _ DF = _ F. _ DF = _ F D. _ DF = _ F. _ F = _ DF rian 5.6 ft 4.2 ft Tree 42.3 ft D F Hougton Mifflin Harcourt Publising ompany Moule Lesson 3

9 H.O.T. Focus on Higer Orer Tinking 20. ritique Reasoning Jesse an Kyle are iking. Jesse is carrying a walking stick. Tey spot a tall tree an use te walking stick as a vertical marker to create similar triangles an measure te tree inirectly. Later in te ay tey come upon a rock formation. Tey measure te rock formation s saow an again want to use similar triangles to measure its eigt inirectly. Kyle wants to use te saow lengt tey measure earlier for te stick. Jesse says tey soul measure it again. Wo o you tink is rigt? 21. rror nalysis ny wants to fin te istance across a river. He locate points as sown in te figure an ten use similar triangles to fin tat = feet. How can you tell witout calculating tat e must be wrong? Tell wat you tink e i wrong an correct is error. 147 ft 300 ft 200 ft D Hougton Mifflin Harcourt Publising ompany Moule Lesson 3

10 Lesson Performance Task roun 240.., te Greek astronomer ratostenes was resiing in lexanria, gypt. He believe tat te art was sperical an conceive of an experiment to measure its circumference. t noon in te town of Syene, te sun was irectly overea. stick stuck vertically in te groun cast no saow. t te same moment in lexanria, 490 miles from Syene, a vertical stick cast a saow tat veere 7.2 from te vertical. 1. Refer to te iagram. xplain wy ratostenes reasone tat te angle at te center of te art tat intercepte a 490-mile arc measure 7.2 egrees. 2. alculate te circumference of te art using ratostenes s figures. xplain ow you got your answer. 3. alculate te raius of te art using ratostenes s figures. 4. Te accepte circumference of te art toay is 24,901 miles. alculate te percent error in ratostenes s calculations. 7.2 lexanria 7.2 Syene saow 490 mi no saow Hougton Mifflin Harcourt Publising ompany Moule Lesson 3