To identify and apply similar polygons. two polygons are similar and to find unknown side lengths of similar figures. Key Concept Similar Polygons

Transcription

1 7-2 Similar olgons ontent Standard.S.5 Use... similarit criteria for triangles to solve problems and to prove relationships in geometric figures. Objective o identif and appl similar polgons an ou use what ou ve learned before about ratios to help ou solve this problem? movie theater screen is in the shape of a rectangle 45 ft wide b 25 ft high. Which of the screen formats at the right do ou think would show the most complete scene from a movie shown on the theater screen? plain. MHMI IS Similar figures have the same shape but not necessaril the same size. You can abbreviate is similar to with the smbol. YMI I I S S I namic ctivit Similar olgons ssential Understanding You can use ratios and proportions to decide whether two polgons are similar and to find unknown side lengths of similar figures. esson ocabular similar s figures similar polgons etended proportion scale factor scale drawing scale efine wo polgons are similar polgons if corresponding angles are congruent and if the lengths of corresponding sides are proportional. Ke oncept Similar olgons iagram HIJ H I J Smbols H I J H HI IJ J You write a similarit statement with corresponding vertices in order, just as ou write a congruence statement. When three or more ratios are equal, ou can write an etended proportion. he proportion H HI IJ J is an etended proportion. scale factor is the ratio of corresponding linear measurements of two similar figures. he ratio of the lengths of corresponding 20 Y sides and YZ, or more simpl stated, the ratio of corresponding sides, is 20 YZ 5 2. So the scale factor X of to XYZ is or 5 : 2. XYZ Z 440 hapter 7 Similarit

2 How can ou use the similarit statement to write ratios of corresponding sides? Use the order of the sides in the similarit statement. M corresponds to S, so M S is a ratio of corresponding sides. roblem 1 M S ot It? Understanding Similarit What are the pairs of congruent angles? M S,, and What is the etended proportion for the ratios of corresponding sides? M S M S 1. HJK. a. What are the pairs of congruent angles? b. What is the etended proportion for the ratios of the lengths of corresponding sides? M S How do ou identif corresponding sides? he included side between a pair of angles of one polgon corresponds to the included side between the corresponding pair of congruent angles of another polgon. roblem 2 etermining Similarit re the polgons similar? If the are, write a similarit statement and give the scale factor. JKM and UW J K U Step 1 Identif pairs of congruent angles. M W J, K U,, and M W Step 2 ompare the ratios of corresponding sides. 24 JK U 2 1 M 24 W 14 7 K 24 U JM W 1 1 orresponding sides are not proportional, so the polgons are not similar. and Step 1 Step 2 Identif pairs of congruent angles.,, and ompare the ratios of corresponding sides Yes; and the scale factor is 4 5 or ot It? 2. re the polgons similar? If the are, write a similarit statement and give the scale factor. a. K 10 W 20 X b. 1 S 1 U M Z Y 1 esson 7-2 Similar olgons 441

3 roblem 3 Using Similar olgons an ou rel on the diagram alone to set up the proportion? o, ou need to use the similarit statement to identif corresponding sides in order to write ratios that are equal. lgebra. What is the value of? orresponding sides of similar polgons are proportional. Substitute. ross roducts ropert ivide each side b. he value of is 5. he correct answer is. ot It? 3. Use the diagram in roblem 3. What is the value of? You can t solve the problem until ou know which dimension fills the space first. roblem 4 Using Similarit esign Your class is making a rectangular poster for a rall. he poster s design is in. high b 10 in. wide. he space allowed for the poster is 4 ft high b ft wide. What are the dimensions of the largest poster that will fit in the space? Step 1 Step 2 etermine whether the height or width will fill the space first. Height: 4 ft 4 in. Width: ft in. 4 in. in. in. 10 in.. he design can be enlarged at most times. he greatest height is 4 in., so find the width orresponding sides of similar polgons are proportional. 40 ross roducts ropert 0 ivide each side b. he largest poster is 4 in. b 0 in. or 4 ft b 2 3 ft. ot It? 4. Use the same poster design in roblem 4. What are the dimensions of the largest complete poster that will fit in a space 3 ft high b 4 ft wide? 442 hapter 7 Similarit

4 In a scale drawing, all lengths are proportional to their corresponding actual lengths. he scale is the ratio that compares each length in the scale drawing to the actual length. he lengths used in a scale can be in different units. or eample, a scale might be written as 1 cm to 50 km, 1 in. 100 mi, or 1 in. : 10 ft. You can use proportions to find the actual dimensions represented in a scale drawing. roblem 5 Using a Scale rawing SM esign he diagram shows a scale drawing of the olden ate ridge in San rancisco. he distance between the two towers is the main span. What is the actual length of the main span of the bridge? Wh is it helpful to use a scale in different units? 1 cm 200 m in the same units would be 1 cm 20,000 cm. When solving the problem, is easier to work with 1 than 20,000. he length of the main span in the scale drawing is.4 cm. et s represent the main span of the bridge. Use the scale to set up a proportion s ot It? s 0 length in drawing (cm) actual length (m) ross roducts ropert he actual length of the main span of the bridge is 0 m. 5. a. Use the scale drawing in roblem 5. What is the actual height of the towers above the roadwa? b. easoning he Space eedle in Seattle is 05 ft tall. classmate wants to make a scale drawing of the Space eedle on an 1 2 in. b-11 in. sheet of paper. He decides to use the scale 1 in. 50 ft. Is this a reasonable scale? plain. esson 7-2 Similar olgons 443

5 esson heck o ou know HOW? J WHYX. omplete each statement YX WX 3. re the polgons similar? If the are, write a similarit statement and give the scale factor. H H M. What is the value of? M H 20 1 Q o ou US? 5. ocabular What does the scale on a scale drawing indicate?. rror nalsis he polgons at the right are similar. Which similarit statement is not correct? plain.. U QU. U QU 7. easoning Is similarit refleive? ransitive? Smmetric? Justif our reasoning.. he triangles at the right are similar. What are three similarit statements for the triangles? MHMI IS U Q S ractice and roblem-solving ercises MHMI IS ractice ist the pairs of congruent angles and the etended proportion that relates the corresponding sides for the similar polgons. See roblem 1.. S 10. W 11. KM H W K H S M etermine whether the polgons are similar. If so, write a similarit statement and give the scale factor. If not, eplain K 1 J See roblem Q hapter 7 Similarit

6 . M H 10 U 1. H 1 14 I S J lgebra he polgons are similar. ind the value of each variable. See roblem z SM SM 21. Web age esign he space allowed for the mascot on a school s Web page is 0 piels wide b 0 piels high. Its digital image is 500 piels wide b 375 piels high. What is the largest image of the mascot that will fit on the Web page? 22. rt he design for a mural is 1 in. wide and in. high. What are the dimensions of the largest possible complete mural that can be painted on a wall 24 ft wide b 14 ft high? 23. rchitecture You want to make a scale drawing of ew York it s mpire State uilding using the scale 1 in. 250 ft. If the building is 50 ft tall, how tall should ou make the building in our scale drawing? 24. artograph cartographer is making a map of ennslvania. She uses the scale 1 in. 10 mi. he actual distance between Harrisburg and hiladelphia is about 5 mi. How far apart should she place the two cities on the map? See roblem 4. See roblem 5. ppl In the diagram below, HKM. ind each of the following. 25. the scale factor of HKM to 2. mk 27. MH 2. MK lags compan produces a standard-size U.S. flag that is 3 ft b 5 ft. he compan also produces a giant-size flag that is similar to the standard-size flag. If the shorter side of the giant-size flag is 3 ft, what is the length of its longer side? 31. a. oordinate eometr What are the measures of,,,,,, and? plain. b. What are the lengths of,,,,,,, and? c. Is similar to? Justif our answer M 5 30 H 1 K 2 O 1 4 esson 7-2 Similar olgons 445

7 32. hink bout a lan he avis famil is planning to drive from San ntonio to Houston. bout how far will the have to drive? How can ou find the distance between the two cities on the map? What proportion can ou set up to solve the problem? el io ustin Houston San ntonio alveston 33. easoning wo polgons have corresponding side lengths that are proportional. an ou conclude that the polgons are similar? Justif our reasoning. 34. Writing plain wh two congruent figures must also be similar. Include scale factor in our eplanation. 35. JK and S are similar. he scale factor of JK to S is 3 1. What is the scale factor of S to JK? Scale 1 cm : 1 km aredo 3. Open-nded raw and label two different similar quadrilaterals. Write a similarit statement for each and give the scale factor. orpus hristi rownsville lgebra ind the value of. ive the scale factor of the polgons. 37. WJ Q 3. KM W J 5 M Q K Sports hoose a scale and make a scale drawing of each rectangular plaing surface. 3. soccer field is 110 d b 0 d. 40. volleball court is 0 ft b 30 ft. 41. tennis court is 7 ft b 3 ft. 42. football field is 30 ft b 10 ft. etermine whether each statement is alwas, sometimes, or never true. 43. n two regular pentagons are similar. 44. heagon and a triangle are similar. 45. square and a rhombus are similar. 4. wo similar rectangles are congruent. SM 47. rchitecture he scale drawing at the right is part of a floor plan for a home. he scale is 1 cm 10 ft. What are the actual dimensions of the famil room? ining Kitchen hallenge 4. he lengths of the sides of a triangle are in the etended ratio he perimeter of the triangle is 54 in. a. he length of the shortest side of a similar triangle is 1 in. What are the lengths of the other two sides of this triangle? b. ompare the ratio of the perimeters of the two triangles to their scale factor. What do ou notice? amil room Master bedroom 44 hapter 7 Similarit

8 4. In rectangle, 2 3. In rectangle JW, J J 2 3. Show that JW. 50. rove the following statement: If and HK, then HK. Standardized est rep S/ 51. QS JKM with a scale factor of 4 3. Q cm. What is the value of K? cm cm cm 24 cm 52. Which of the following is O a propert of an isosceles trapezoid? he base angles are congruent. he diagonals are perpendicular. he legs are congruent. he diagonals are congruent. Short esponse 53. In the diagram at the right, what is m1? high school communit-action club plans to build a circular pla area in a cit park. he club members need to bu materials to enclose the area and sand to fill the area. or a -ft-diameter pla area, what will be the circumference and area rounded to the nearest hundredth? Mied eview If 7, complete each statement using the properties of proportions. See esson Use the diagram for ercises 5 1. See esson ame the isosceles triangles in the figure m 3 42 et ead! o prepare for esson 7-3, do ercises 2 4. How can ou prove that the triangles are congruent? See essons 4-2 and esson 7-2 Similar olgons 447