7.2 Similar Polygons - Carl Schurz High School
Page 1 of 77.2GoalSimilar PolygonsIn geometry, two figures that have the same shape are called similar.Identify similar polygons.Key Words similar polygons scale factorTwo polygons are similar polygons if corresponding anglesare congruent and corresponding side lengths are proportional.In the diagram, ABCD is similar to EFGH. The symbol S indicatessimilarity. So, you can write ABCD S EFGH. When you refer to similarpolygons, list their corresponding vertices in the same order.CGBFHEDAABCD S EFGHCorresponding AnglesRatios of Corresponding SidesaA c aEaB c aFaC c aGaD c aHEFFGGHHE ABBCCDDAStudent HelpVOCABULARY TIPTPRQ S TSTU iscalled a similaritystatement.EXAMPLEUse Similarity Statements1TPRQ S TSTU.Ta. List all pairs of congruent angles.16Rb. Write the ratios of thecorresponding sides in astatement of proportionality.82010SPc. Check that the ratios of12UP15corresponding sides are equal.Solutiona. aP c aS, aR c aT, and aQ c aU.STPRTURQUSQPSTPR8104 TU5 RQb. 162045USQP121545c. , , and .45The ratios of corresponding sides are all equal to .7.2Similar Polygons365
Page 2 of 7Scale Factor If two polygons are similar, then the ratio of the lengthsof two corresponding sides is called the scale factor .CGBFHEDAEFABFGBCGHCDHEDAscale factor of EFGH to ABCD IStudent HelpDetermine Whether Polygons are Similar2EXAMPLEICLASSZONE.COMDetermine whether the triangles are similar. If they are similar,write a similarity statement and find the scale factor of Figure Bto Figure A.MORE EXAMPLESMore examples atclasszone.comH9K12AG1512B16JM20SolutionL1 Check whether the corresponding angles are congruent. From the diagram, you can see that aG c aM, aH c aK, andaJ c aL. Therefore, the corresponding angles are congruent.2 Check whether the corresponding side lengths are proportional. 4MK1212 3 3GH99 3All three ratios are equal, sothe corresponding side lengthsare proportional.KL1616 44 HJ1212 434LM2020 5 3JG1515 5ANSWER By definition, the triangles are similar. TGHJ S T MKL.43The scale factor of Figure B to Figure A is .Determine Whether Polygons are SimilarDetermine whether the polygons are similar. If they are similar, write asimilarity statement and find the scale factor of Figure B to Figure A.E1. Y2. NAX89ZB18FSimilarityA129PD12Chapter 7S126366PRVTB106U
Page 3 of 7EXAMPLE3Use Similar PolygonsT RST S T GHJ.Find the value of x.HS10Student HelpSTUDY TIPYou can find x usingother proportions thatinclude TR. Anotherproportion you can useSTHJTRJGis .Rx15128TG9JSolutionBecause the triangles are similar, the corresponding side lengthsare proportional. To find the value of x, you can use the followingproportion.JGGH TRRS159 10xWrite proportion.Substitute 15 for GH, 10 for RS,9 for JG, and x for TR.15 p x 10 p 9Cross product property15x 90Multiply.15x90 1515Divide each side by 15.x 6EXAMPLESimplify.4Perimeters of Similar PolygonsThe outlines of a pool and the patio around the pool aresimilar rectangles.a. Find the ratio of the lengthof the patio to the length ofthe pool.16 ftb. Find the ratio of the24 f t32 ftperimeter of the patio tothe perimeter of the pool.48 ftSolutiona. The ratio of the length of the patio to the length of the pool islength of patio48 feet48 163 .length of pool32 feet32 162b. The perimeter of the patio is 2(24) 2(48) 144 feet.The perimeter of the pool is 2(16) 2(32) 96 feet.The ratio of the perimeter of the patio to the perimeter ofthe pool isperimeter of patio144 feet144 483 .96 feet96 482perimeter of pool7.2Similar Polygons367
Page 4 of 7In Example 4 on the previous page, notice that the ratio of theperimeters of the similar figures is equal to the ratio of the sidelengths. This observation is generalized in the following theorem.THEOREM 7.1Perimeters of Similar PolygonsWordsIf two polygons are similar, then theratio of their perimeters is equalto the ratio of their correspondingside lengths.SymbolsDACBFDE EF FDAB BC CADEABEEFBCFDCAIf T ABC S T DEF, then .Use Similar PolygonsIn the diagram, T PQR S T STU.P3. Find the value of x.Sx14T4. Find the ratio of the perimeter of T STUto the perimeter of T PQR.P978UR167.2 ExercisesGuided PracticeVocabulary Check1. If two triangles are congruent, must they be similar? Explain.2. If two triangles are similar, must they be congruent? Explain.Skill CheckIn Exercises 3–6, T ABC S T LMN.B3. List all pairs of congruent angles.M4. Write the ratios of the correspondingsides in a statement of proportionality.28xA205. Find the scale factor of T LMN to T ABC.21CL2715N6. Find the value of x.F7. Are the two rectangles shownat the right similar? Explainyour reasoning.368Chapter 7Similarity10P4PSG7RJ15H
Page 5 of 7Practice and ApplicationsExtra PracticeSee p. 687.Using Similarity Statements List all pairs of congruent angles.Then write the ratios of the corresponding sides in a statementof proportionality.8. T PQR S T DEF9. ABCDE S QRSTUAPPREFEPCBDRDSUT10. Error Analysis TFGH S TJKL. A student was asked to list allpairs of congruent angles and write the ratios of the correspondingsides in a statement of proportionality. Copy the diagram andcorrect the student’s errors.aF c aJ, aG c aL, aH c aKLKFKLFGFH GHJKJLGJHDetermining Similarity Determine whether the polygons are similar.If they are similar, write a similarity statement and find the scalefactor of Figure B to Figure A.11.1520AG12. PEH246B 18DK10AN2012M8B9F12JL16J13. JKA44LE14.FA5BSS Z10 Z10VYMHHomework HelpExample 1: Exs. 8–10,22, 23Example 2: Exs. 11–16,29Example 3: Exs. 17–21,24, 25Example 4: Exs. 26, 27U15. TUA5F 16. X6VBH67XY36RSB4126.75WTBGE48W4.58ATGZ97.2Similar PolygonsY369
Page 6 of 7Using Similar Polygons The two polygons are similar. Find the valuesof x and y.P17. JxPL19. J18612GD12xLy6LG 4M9JNR8KxP16PCareers21. Mural Alejandro Romero created the mural, Chicago Federation ofLabor, by enlarging the 56 in. by 21 in. sketch shown below. Romeroused a scale factor of about 3.5. What are the dimensions of themural in inches? In feet?ARTIST Some artists creatework for public, educational,or commercial purposes.Alejandro Romero, above,creates large murals that aredisplayed in public buildingsCareer LinksUsing Similar Polygons In Exercises 22–26, use the diagram below,where FGHJ S KLMN.22. Find maN.23. Find maF.CLASSZONE.COM24. Find the value of x.25. Find the value of y.KF xGJ12980 Ly10HN18M26. Find the ratio of the perimeter of KLMN to the perimeter of FGHJ.27. Perimeters of Similar Polygons TQRS is similar to T XYZ. Theratio of one side of T XYZ to the corresponding side of TQRS is6 : 7. What is the ratio of the perimeter of T XYZ to the perimeterof TQRS?28. Challenge T JKL is similar to T STU. The ratio of ST to JK is 5 to 2.The perimeter of TSTU is 35 feet. Find the perimeter of T JKL.370Chapter 18.
Page 7 of 729. Television Screens The aspect ratio of a television screen is thelength-to-width ratio of the screen. A standard definition television hasan aspect ratio of 4 : 3. A high definition projection television has anaspect ratio of 16 : 9. Are the television screens similar rectangles?Technology20 in.15 in.25 in.EXAMPLELogical ReasoningAre two isosceles trapezoids always, sometimes, or never similar?SolutionTELEVISION SCREENShave sizes that are based onthe length of their diagonal.The aspect ratio of a 25-inchstandard definition televisionscreen is 20 in. : 15 in., or 4 : 3.3It is possible to sketch twosimilar isosceles trapezoids.323226It is also possible to sketchtwo isosceles trapezoids thatare not similar.ANSWER472 50 Therefore, two isosceles trapezoids are sometimes similar.Logical Reasoning Are the polygons always, sometimes, ornever similar?Standardized TestPracticeMixed Review30. Two isosceles triangles31. Two rhombuses32. Two equilateral triangles33. A right and an isosceles triangle34. Multiple Choice T DEF S T MNP. Which statement may be false?A aE c aNB aP c aDC MNNP DEEFD MNPM DEFDShowing Triangles are Congruent Does the diagram give enoughinformation to show that the triangles are congruent? If so, statethe postulate or theorem you would use. (Lessons 5.3, 5.4)35.P36. PASBCTFAlgebra SkillsLRDJ37.MKUEWriting Equivalent Fractions Write two equivalent fractions.(Skills Review, p. 656)1338. 2539. 944740. 41. 7.2Similar Polygons371
Feb 29, 2012 · Use Similar Polygons Perimeters of Similar Polygons Words If two polygons are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths. Symbols If TABC STDEF, then A D B E B E C F C FD A A D B E B E C F. C FD A THEOREM 7.1 F D E C A B 14
-Bega.recessed.blk 1240 260 polygons-Bega.recessed.blk 2083 89 polygons-Bega.recessed.blk 2216 109 polygons-Bega.recessed.blk 2286p 88 polygons-Bega.recessed.blk 2289p 69 polygons-Bega.recessed.blk 2414 6 polygons-Bega.surface.blk 2457p 302 polygons-Bega.surface.blk 2491p 17 polygons-Bega.surface.blk 2541p 331 polygons-Bega.surface.blk 2614p .
Similar Polygons When two polygons have the same shape and only differ in size, we say they are similar polygons. These two pentagons are similar. More formally, two polygons are similar if and only if there is a one-to-one correspondence between their vertices suc
Find corresponding lengths in similar polygons. Find perimeters and areas of similar polygons. Decide whether polygons are similar. Using Similarity Statements Recall from Section 4.6 that two geometric fi gures are similar fi gures if and only if there is a similarity transformation that maps one fi gure onto the other. Using Similarity Statements
draw similar polygons solve problems using the properties of similar polygons Similar Polygons The single star in a Lakota star quilt is made from fabric cut into diamond shapes a
Feb 29, 2012 · and SAS Similarity Theorems. Key Words similar polygons p. 365 The triangles in the Navajo rug look similar. To show that they are similar, you can use the definition of similar polygons or the AA Similarity Postulate. In this lesson, you will learn two new methods to show that two triangles are similar. 7.4 Showing Triangles are Similar .
7-1 Ratios in Similar Polygons Lesson Quiz: Part I 1. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 2. The ratio of a model sailboat’s dimensions to the actual boat’s dimensions is . If the length of the model is 10 inches, wh
Similar Polygons Polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. The symbol: means is similar to AR Proportionality Statement: AB BC CD DE AE RS ST TU UV RV The ratio of the lengths of two corresponding sides o
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