Proportions and Similar Figures. Simplify each ratio = Investigation: Proportions in Triangles

Transcription

1 - Lesson Preview What You ll Learn OJTIV OJTIV To find missing measures of similar figures To use similar figures when measuring indirectly... nd Why To apply proportions when finding distances represented on maps, as in ample Proportions and Similar igures heck Skills You ll Need (or help, go to the Skills andbook and Lesson -.) Simplify each ratio Solve each proportion.. 7 y. w z a 0 0. n n New Vocabulary similar figures scale drawing scale -. Plan Lesson Preview heck Skills You ll Need Simplifying ractions Skills andbook: p. 7 ample, ercises Ratio and Proportion Lesson -: ample ercises tra Practice, p. 70 OJTIV Part Similar Using Similar igures igures Interactive lesson includes instant self-check, tutorials, and activities. Lesson Resources Teaching Resources Practice, Reteaching, nrichment heckpoint Quiz Investigation: Proportions in Triangles The figure below shows n and n.. cm, 7.6 cm,.6 cm,. cm,.6 cm,. cm Reaching ll Students Practice Workbook - Spanish Practice Workbook - Reading and Math Literacy Spanish Reading & Literacy Spanish heckpoint Quiz asic lgebra Planning Guide - Presentation ssistant Plus! Transparencies heck Skills You ll Need - dditional amples - Student dition nswers - Lesson Quiz - P Presentation Pro - Real-World onnection The triangles in the quilt are the same shape, so they are similar.. Measure,,,,, and using a metric ruler. See above.. ind each ratio. a. 0.6 b. 0.6 c Tell whether each statement is true. a. true b. true c. true. Using the lengths you have measured, write two ratios that equal,. Ongoing ssessment and Intervention efore the Lesson iagnose prerequisite skills using: heck Skills You ll Need uring the Lesson Monitor progress using: heck Understanding dditional amples Standardized Test Prep Lesson - Proportions and Similar igures fter the Lesson ssess knowledge using: Lesson Quiz omputer Test Generator hapter heckpoint (p. ) heckpoint Quiz omputer Test Generator Technology Resource Pro -ROM omputer Test Generator Prentice all Presentation Pro Student Site Teacher Web ode: aek-00 Self-grading Lesson Quiz Teacher enter Lesson Planner Resources Plus

2 . Teach Math ackground Not only the corresponding sides, but also the medians, altitudes, and other corresponding lengths of similar figures are all in the same proportion. This relationship can be used to measure indirectly the height of a flagpole or the distance across a body of water. OJTIV Teaching Notes Investigation (Optional) Visual Learners raw a scalene triangle for the overhead projector. Project the image onto the chalkboard. Trace the triangle on the board. Move the projector closer to the board so that the image gets smaller. Trace the triangle again. sk students to compare the triangles. same shape, different sizes XMPL nglish Learners Some students may not understand the term corresponding. ave a student look up the definition in a dictionary and read it aloud to the class. In figures, corresponding parts are in matching places. or eample, corresponds to G because they are between pairs of congruent angles in similar triangles. dditional amples Need elp? ongruent angles have equal measures. The symbol > means is congruent to. In the diagram below, n and ng are similar. Similar figures have the same shape but not necessarily the same size. The symbol, means is similar to. In similar triangles, corresponding angles are congruent and corresponding sides are in proportion. The order of the letters indicates the corresponding angles. If n, ng, then the following is true. XMPL inding the Length of a Side Geometry In the figure below, n, n. ind. & > & & > &G & > & Relate efine Let. Write a proportion comparing the lengths of corresponding sides. Write 0 Substitute for, 0 for, for, and for. 0() Write cross products. 0 ivide each side by. Simplify. is cm. 0 cm cm G G 0 0 cm G In the figure below,. ind. mm heck Understanding In the figure below, ng, nklm. ind LM. cm K mm cm cm mm 6 mm G 0 cm L M 0 hapter Solving and pplying Proportions Reaching ll Students 0 elow Level Suggest that before students write a proportion involving similar figures, they point to and say to themselves the names of the corresponding sides. dvanced Learners hallenge students to write as many proportions as possible that can be used to solve the problem in ample. nglish Learners See note on page 0. Inclusion See note on page.

3 OJTIV Part Indirect Measurement and Scale rawings heck Understanding You can use proportions to find the dimensions of objects that are difficult to measure directly. XMPL pplying Similarity Indirect Measurement tree casts a shadow 7. ft long. woman ft tall casts a shadow ft long. The triangle shown for the tree and its shadow is similar to the triangle shown for the woman and her shadow. ow tall is the tree? 7. 7.? 7. orresponding sides of similar figures are in proportion. Write cross products. Simplify.. ivide each side by. The tree is. ft tall. ft 7. ft a. tree casts a 6-ft shadow. boy standing nearby casts a -ft shadow. is height is. ft. ow tall is the tree?.7 ft b. house casts a 6-ft shadow. girl standing nearby casts a 7.-ft shadow. er height is. ft. What is the height of the house? ft OJTIV Teaching Notes onnection to Geometry Some students may wonder why the two triangles are similar. One way to prove triangles are similar is the ngle-ngle Similarity Theorem. If you can prove two pairs of corresponding angles of two triangles are congruent, then the two triangles are similar. The angles formed by the sun s rays and the standing objects are the same. ach object is standing at a right angle with the ground. XMPL XMPL Inclusion Some students may have difficulty measuring because of disabilities. Let students work in pairs. dditional amples flagpole casts a shadow 0 feet long. man 6 ft tall casts a shadow 7 feet long. ow tall is the flagpole? 6 ft Valkaria TLNTI LO RI Grant ON N W Micco Roseland Sebastian Wabasso Gifford S scale drawing is an enlarged or reduced drawing that is similar to an actual object or place. loor plans, blueprints, and maps are all eamples of scale drawings. The ratio of a distance in the drawing to the corresponding actual distance is the scale of the drawing. XMPL inding istances on Maps The scale of the map at the left is inchi0 miles. pproimately how far is it from Valkaria to Wabasso? Map distance.7 in. Measure the map distance. map S.7 d map actual S 0 d d actual Write a proportion.? d 0?.7 Write cross products. d 7. Simplify. Wabasso is about 7. mi from Valkaria. The scale of a map is inch 0 miles. The map distance from Valkaria to Gifford is. inches. ow far is the actual distance?. mi losure sk: ow can you use proportions to find a distance that is difficult to measure? You can use the shadow and height of a figure that you can measure and the shadow of the figure you cannot measure to write a proportion. Then you can solve the proportion. heck Understanding a. On the map above, measure the map distance from Grant to Gifford. ind the actual distance. about mi b. ritical Thinking If another map showed the distance from Valkaria to Wabasso but had a scale of inch: miles, what would the map distance be between the two locations?. in. Lesson - Proportions and Similar igures

4 . Practice ssignment Guide Objective ore,,, tension 6 Objective ore 0,, tension Standardized Test Prep 7 0 Mied Review Visual Learners ercises Suggest to students that they trace the figures onto their papers. They can then use colored pencils or markers to color code the corresponding parts. nrichment - Reteaching - Practice - Name lass ate Practice - ach pair of figures is similar. ind the length of Proportions and Similar igures 7 6. XRISS Practice and Problem Solving Practice by ample ample (page 0). O PQ, O QR, O RP; l OlP, l OlQ, l OlR. O J, O JK, O K; l Ol, l OlJ, l OlK ample (page ) The figures in each pair are similar. Identify the corresponding sides and angles.. P Q.. See left. PQR The figures in each pair are similar. ind the missing length.. ft... cm ft 0 cm ft y. 0 in. 0 in m cm. cm ft 7 cm in. cm The child in the figure is ft tall. R in. or more practice, see tra Practice. k 6 ft m k JK cm 6 m ft 7. ft K J m 6 ft Pearson ducation, Inc. ll rights reserved. 7.. Use a proportion to solve.. n is similar to nxyz. The length is 0. The length is 7. ind the length XY if the length YZ is. 0. Marty has a scale model of a car. The scale is in. : in. If the model is 6.7 in. long, how long is the actual car?. blueprint scale is in. : ft. The width of a building is ft. What is the width of the building on the blueprint?. ngie is using similar triangles to find the height of a tree. stick that is ft tall casts a shadow that is ft long. The tree casts a shadow that is ft long. ow tall is the tree?. n is similar to nxyz. The length is 0. The length is 6. What is the length XZ if the length YZ is?. map has a scale of in. : mi. Two cities are 7 mi apart. ow far apart are they on the map? lgebra hapter Lesson - Practice. ow tall is the tree?. ft 60 in. ft 0. The cat casts an -in. shadow. ow tall is the cat? in. ample (page ) The scale of a map is in.i7. mi. ind the actual distance corresponding to each map distance.. in. 7. mi.. in.. mi..6 in.. mi. 0 in. 0 mi hapter Solving and pplying Proportions

5 Lincoln SL in. : 6 mi San Paulo uncanville pply Your Skills a. nswers may vary. Sample: GK and RQ are not corresponding sides. 0c. Yes, the ratio of the sides is equal to the ratio of perimeters in similar figures. d. m, m. a. Use a ruler and the map at the left. ind the distance from each town to the others. See margin. b. student lives halfway between Lincoln and San Paulo and takes the shortest route to school in uncanville. ow far does the student travel each day to school? 6 mi 6. The actual distance between two towns is km. Suppose you measure the distance on your map and find that it is. cm. What is the scale of your map? cm : km Using each of the following scales, find the dimensions in a blueprint of an ft-by- ft room. 7. in.i ft. in.i ft. in.i ft 0. in.i. ft in. by 6 in. in. by in. in. by in.. in. by. in.. Two rectangles are similar. The first is in. wide and in. long. The second is in. wide. ind the length of the second rectangle..7 in.. rchitecture blueprint scale is in.i ft. On the plan, the room measures. in. by in. What are the actual dimensions of the room?. ft by 7 ft. rror nalysis The two figures Q R are similar. Robert uses the 7 m G GK. m proportion PQ RQ to find RQ. G m P a. What is Robert s error? 7 m b. What proportion should he L. m G L K m N have used? PQ RQ rchitecture -in. length in the scale drawing represents an actual length of ft.. What is the scale of the drawing? in. : ft. What are the actual dimensions of the kitchen? ft by ft 6. ind the actual width of the doorways that lead into the kitchen and the dining room. ft 7. ind the actual area of the dining room. 6 ft. an a table 7 ft long and ft wide fit into the narrower section of the dining room? plain your answer. yes; because it is 6 ft wide and ft long. Two rectangles are similar. One is cm by cm. The longer side of the second rectangle is greater than twice its shorter side. ind its length and width. long by 0 cm wide 0. Geometry Rectangle is similar to rectangle KLMN. a. What is the width w of rectangle KLMN? 6 m b. What is the perimeter of each rectangle? 6 m, m m c. Is the ratio of the perimeters of the rectangles (smallilarge) equal to the m ratio of corresponding sides? plain. w d. What is the area of each rectangle? K L e. ritical Thinking ind the ratio of the areas (smallilarge). plain how the ratio of the areas is m related to the ratio of the corresponding sides. N M nswers may vary. Sample: The area ratio is the square of the side ratio.. Open-nded Give some eamples of similar figures found in everyday life. See margin. rror Prevention ercises, 0 Students may have difficulty writing the proportion correctly. ave students write labels on each term as they write the proportion. or eample: in. (child s shadow) ft (tree s shadow) y seeing the matching words, students can determine if they wrote corresponding measurements in the same ratio. ercise 6 Let students know that a map scale is an instance of a unit rate. areers ercise rchitects draw buildings, houses, and landscapes to scale. These scale drawings must be eact so that the builders know eactly what size the building materials must be to fit together properly. ercise Remind students that only corresponding sides of similar figures should be used to write proportions. pages ft (child s height) (tree s height) ercises a. Lincoln to San Paulo 6 mi Lincoln to uncanville 6 mi San Paulo to uncanville mi Lesson - Proportions and Similar igures

6 . ssess Lesson Quiz -. In the figure below,. ind. about.7 cm cm 7. cm. boy who is. feet tall casts a shadow that is. feet long. The tree net to him casts a shadow that is feet long. ow tall is the tree? ft. The scale on a map is in. 0 mi. What is the actual distance between two towns that are. inches apart on the map? 70 mi Problem Solving int or ercise, drawing a diagram can help you understand the problem. hallenge. a. Writing re the two cubes similar? plain your answer. b. plain how the ratio of volumes (small : large) is related to the ratio of their sides (small : large). c. If the ratio of the sides of the two cubes is :, what is the ratio of their volumes? a-c See margin.. Geometry The perimeter of a triangle with sides a, b, and c is cm. Side a is cm longer than side b. The ratio of the lengths of sides b and c is :. What are the lengths of the three sides of the triangle? a, b 6, c 0. The state of labama is about mi long and 0 mi wide. What scale would you use to draw a map of labama on an in.-by- in. paper to make the map as large as possible? about in. :.7 mi. stronomy You can block out the moon by holding a coin up at a distance from your eye that is 0 times the diameter of the coin. Using similar figures, coin diameter coin distance moon diameter moon distance. The moon is roughly 60 kilometers in diameter. ow far away is it? 00,00 km Not drawn to scale. istance to coin iameter of coin istance to moon cm cm cm iameter of moon lternative ssessment Group students in pairs. Instruct them to use shadows and proportions to find the height of objects they cannot measure, such as a building, a tree, or a flagpole. 6. Geometry In the figure at the right, #, #. a. Substitute values from the diagram into the following proportion. 7 (int: +.) b. Solve the proportion for.. c. ind the length of.. in. d. What is the area of #?. in. in. 7 in. Standardized Test Prep in. Resources or additional practice with a variety of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 6 Test-Taking Strategies with Transparencies onnection to Geometry ercise 7 Sometimes it is confusing to look at the figures. You can use the similarity statement to determine if a proportion is correct. ll s will correspond with all X s since they are the first letter written for each triangle in the similarity statement. This is also true for all s and Y s, and for all s and Z s. Standardized Standardized Test Prep Test Prep Multiple hoice hapter Solving and pplying Proportions pages ercises. nswers may vary. Sample: doll house to regular house, model car to real car 7. In the figure at the right, n, nxyz. Which proportion is incorrect?. XY XZ. XY XZ. YZ XZ. XZ YZ a. Yes; the sides are proportional. b. : ; : ; answers may vary. Sample: The X volume ratio is the cube of the side ratio. c. 7 : Z Y

7 Short Response tended Response Take It to the NT Online lesson quiz at Web ode: aea-00. map of Kentucky is drawn with a scale of cmi km. The map distance between Louisville and owling Green is. cm. Which is the best estimate of the actual distance? I.. km G. km. 00 km I. 60 km. You can paint a 6 ft-by- ft rectangular wall using 0. gallon of paint. ow many gallons of paint will you need to cover a 0 ft-by- ft wall? Show your work. See margin. 0. Leonardo da Vinci s famous painting the Mona Lisa measures 77. cm by cm. See margin. a. plain how you know that a 6 cm-by- cm reproduction postcard is NOT similar to the original painting. b. What dimensions would make a postcard similar to the original painting? pproimate to the nearest tenth. Show your work or eplain how you found your answer. hapter heckpoint To check understanding of Lessons - to -: heckpoint Quiz (p. ) Teaching Resources heckpoint Quiz (also in Prentice all ssessment System) Reaching ll Students Reading and Math Literacy Spanish versions available Mied Review Lesson - Lesson - Solve each proportion.... n m v 0 Solve.. b, $ L m.. 7 h, b R m R h S Lessons heckpoint - through Quiz - Lessons - through - Instant self-check quiz online and on -ROM. [] smaller area: 6? 0; 0 ft ; larger area: 0? 0; 0 ft ; (0) omplete the statement days j minutes. 0. Write $ for hours as a unit rate. $6.00/h Solve each proportion y 0. 6 k You are riding your bicycle. It takes you min to go. mi. If you continue traveling at the same rate, how long will it take you to go 7 mi?.6 min The figures in each pair are similar. ind the missing length. 7.. cm. cm cm cm y..-ft child casts a 60-in. shadow. She is standing net to a telephone pole that casts a 0-ft shadow. ow tall is the telephone pole? ft 0. The scale of a map is in. i 0 mi. ind the actual distance if the map distance between two towns is. in. 6 mi Two gallons of paint should cover a 0 ft 0 ft wall. ft 6 ft. ft ft Lesson - Proportions and Similar igures m [] incorrect calculation for one area and proportion solved correctly OR correct area calculations but proportion set up incorrectly 77. 0a. [] 6 u 77.() u 6() 0 u 0 Since the cross products are not equal, the proportion is not true. So the postcard is not similar to the painting. 77. b. OR y The postcard should be cm 6. cm OR. cm 6 cm. [] appropriate methods, but with one computational error OR found only one possible postcard size [] incorrect proportions solved correctly [] correct answer with no work shown