4.2 Using Similar Shapes
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1 LESSON 4.2 Using Similar Sapes Proportionalit 7.5. Generalize te critical attributes of similarit, including ratios witin and between similar sapes. ESSENTIL QUESTION How can ou use similar sapes to find unknown measures Finding Unknown Measures in Similar Triangles Wen ou measure te eigt of a door wit a measuring tape, ou are using direct measurement. Te process of using similar sapes and proportions to find a measure is called indirect measurement. ou can use indirect measurement to measure tings tat are difficult to measure directl, like te eigt of a tree. Mat On te Spot EMPLE Hougton Mifflin Harcourt Publising ompan Image redits: Daniel grill/lam ~ KL. Find te unknown measures. Find te unknown side,. Find cm 12 cm 16 cm K = KL 8 28 = = 2_ 7 6 2_ 7 = 12 6 = 42 cm K corresponds to. L 56 cm Write a proportion using corresponding sides. Side corresponds to side K, and side corresponds to side KL. Substitute te known lengts of te sides. K 28 cm Simplif 8 28 to elp ou find a factor of is a factor of 12. Since 2 times 6 is 12, multipl 7 times 6 to find te value of. = 103 orresponding angles of similar triangles ave equal angle measures. Reflect 1. nalze Relationsips Wat oter proportion could be used to find te value of in te eample Eplain. Lesson
2 OUR TURN Online ssessment and Intervention ~ GH. Find te unknown measures cm 31 6 cm 11.6 cm 5 59 = = m 5.8 cm 5 cm H G 3. H cm s 60 cm 120 cm t G 90 cm s = t = Mat On te Spot Determining Unknown Measures in Similar Four-Sided Sapes ou can use similar sapes and proportions to find measures of sides in rectangles and oter four-sided sapes. EMPLE Mat Talk Matematical Processes Wit similar triangles, ou need to find missing sides and angles. W do ou need to find onl missing sides wit rectangles volleball court is a rectangle tat is similar in sape to an Olmpic-sized pool. Find te widt of te pool. 9 m 18 m Let w = te widt of te pool. 50 m Hougton Mifflin Harcourt Publising ompan 122 Unit = 9 w = 9 w 2 w = 25 Te pool is 25 meters wide. Write a proportion using corresponding sides. Since18 divided b 2 is 9, divide 50 b 2 to find te value of w.
3 OUR TURN 4. Tese rectangular gardens are similar in sape. Find te widt of te smaller garden. 42 ft Online ssessment and Intervention 36 ft 54 ft Using Indirect Measurement Wen ou write a proportion to calculate a measurement, ou can write ratios comparing measures of te same object. For eample, use te ratio formed b te eigt of an object and te lengt of its sadow. EMPL 3 EMPLE 7.5. Mat On te Spot Find te eigt of te totem pole. eigt of person eigt of totem pole Write a proportion. = sadow of person sadow of totem pole 5 = = Substitute known values. Since 4 4 = 16, multipl 5 b 4 to find te value of. = ft 5 ft Hougton Mifflin Harcourt Publising ompan Te totem pole is 20 feet tall. 4 ft OUR TURN 5. student wo is 4 feet tall stands beside a tree. Te tree as a sadow tat is 12 feet long at te same time tat te sadow of te student is 6 feet long. Find te eigt of te tree. 6. potograper is taking a poto of a statue of Paul unan, te legendar giant lumberjack. He measures te lengt of is sadow and te sadow cast b te statue. Find te eigt of te Paul unan statue. 6 ft 4.5 ft 1.5 ft Online ssessment and Intervention Lesson
4 Guided Practice ~ in eac pair. Find te unknown measures (Eample 1) in in. z 25 m 8 m 10 m 14 in. 42 in. d m 15 m t s 36 in in. 7 in. 11 in t 28 in. 44 in. 18 ft a 18 ft 25 ft ft 72 ft 84 Te rectangles in eac pair are similar. Find te unknown measures. (Eample 2) in. 19 in in. 19 in in. Q T 12 cm f R 6 cm S D q 5 cm 7. Samanta wants to find te eigt of a pine tree in er ard. Se measures te eigt of te mailbo at 3 feet and its sadow at 4.8 feet. Ten se measures te sadow of te tree at 56 feet. How tall is te tree (Eample 3) ESSENTIL QUESTION HEK-IN 8. Describe ow ou could use our eigt and a ardstick to determine te unknown eigt of a flagpole. Hougton Mifflin Harcourt Publising ompan 124 Unit 2
5 Name lass Date 4.2 Independent Practice cactus casts a sadow tat is 15 ft long. gate nearb casts a sadow tat is 5 ft long. Find te eigt of te cactus. Online ssessment and Intervention 13. rt n art eibit at a local museum features several similarl saped metal cubes welded togeter to make a sculpture. Te smallest cube as a edge lengt of 6 inces. 3 ft a. Wat are te edge lengts of te oter cubes if te ratios of similarit to te smallest cube are 1.25, 4_ 3, 1.5, 7_ 4, and 2 respectivel 5 ft 15 ft 10. Two ramps modeled wit triangles are similar. Wic side of triangle KO corresponds to _ N Eplain. b. If te artist wanted to add a smaller cube wit an edge lengt wit a ratio of 2_ to te sculpture, wat size would 3 te cube be N K O c. W do ou onl ave to find te lengt of one edge for eac cube Hougton Mifflin Harcourt Publising ompan 11. building wit a eigt of 14 m casts a sadow tat is 16 m long wile a taller building casts a 24 m long sadow. Wat is te eigt of te taller building 12. Katie uses a cop macine to enlarge er rectangular design tat is 6 in. wide and 8 in. long. Te new widt is 10 in. Wat is te new lengt ~ PQR in eac pair. Find te unknown measures cm 8 cm 9 cm 89 s 48 m 64 m 20 cm P 30 m P 77 Q R b 30 cm 58 a 40 m 45 R 35 m Q Lesson
6 16. Two common envelope sizes are 3 1_ 2 in. 6 1_ 2 in. and 4 in. 9 1_ in. re tese 2 envelopes similar Eplain. 17. pair of rectangular baking pans come in a set togeter for $15. One pan is 13 inces b 9 inces and te oter pan is 6 inces b 6 inces. Witout doing an calculations, ow can ou tell tat tese pans are not similar FOUS ON HIGHER ORDER THINKING Work rea 18. Draw onclusions In te similar triangles used in indirect measurement wit te sadows of a flagpole and a person, wic sides of te triangles represent te ras of te sun 19. Make a onjecture Do ou tink it is possible to use indirect measurement wit sadows if te sun is directl overead Eplain. 20. nalze Relationsips osep's parents ave planted two gardens. One is square and as an area of 25 ft 2. Te oter one as two sides equal to 2_ 3 of one side of te square, and te oter two sides equal to 5_ of one side 2 of te square. a. Find te dimensions of te oter garden. Eplain ow ou found our answer. Hougton Mifflin Harcourt Publising ompan b. Find te area of te oter garden. 126 Unit 2
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