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UsingScaleinSimilarTriangles

Scaleisavitalpartofhowsimilarfigureswork.Youcannotaccuratelycomparethetwotriangleswithoutit.Scaletellsyouhowmuchbiggerorsmallerthenewfigurebecame.So,thoughitisabsolutelypossibletodeterminesidemeasureswithoutscale,forthishandout,ourfocuswillbeontwothings:1.determiningthescale,and2.usingthescaletodeterminesidemeasures.

Touseyourscale,youmustfirstsetupyourthreescalefractions.Thereareafewwaystodothis,butthetwooftheeasierwaysareshownbelow:

CreatingScaleFractionsfor△GHK~△NMLOptionA:Listthesidesinorder,andmakethefractions

usingwhat’snexttoeachother.OptionB:putthefirsttriangleontopofthefraction,then

rewriteittwolettersatatime.△GHK~△NML

1sttwoletters:GH~NM…turnedintoafraction:GH/NM

scalefraction:!"!"

lasttwoletters:HK~ML…turnedintoafraction:HK/ML

scalefraction:!"!"

outtwoletters:GK~NL…turnedintoafraction:GK/NL

scalefraction:!"!"

Setthemequaltoeachother:

𝐺𝐻𝑁𝑀

=𝐻𝐾𝑀𝐿

=𝐺𝐾𝑁𝐿

△GHK~△NML

𝐹𝑖𝑟𝑠𝑡 △𝑆𝑒𝑐𝑜𝑛𝑑 △

= △ 𝐺𝐻𝐾△ 𝑁𝑀𝐿

1𝑠𝑡 𝑡𝑤𝑜 = 𝑙𝑎𝑠𝑡 𝑡𝑤𝑜 = 𝑜𝑢𝑡 𝑡𝑤𝑜𝐺𝐻𝐾𝑁𝑀𝐿

= 𝐺𝐻𝐾𝑁𝑀𝐿

= 𝐺𝐻𝐾𝑁𝑀𝐿

𝐺𝐻𝑁𝑀

=𝐻𝐾𝑀𝐿

=𝐺𝐾𝑁𝐿

Createascalefractionequationforeachtriangleset.1.△SLU~△RPS 2.△GQA~△EHI 3.△WXC~ △ZTD

4.△MYK~ △JBR 5.△PVF~△NSZ 6.△LAU~△GHS

Todeterminethescale,pluginthegiveninformation(tothefractionthatgoeswiththatinformation)andsimplify.Example:△SLU~△RPS𝐿𝑈 = 4 & 𝑃𝑆 = 12.Whatisthescale?

𝑆𝐿𝑈𝑅𝑃𝑆

= 𝑆𝐿𝑈𝑅𝑃𝑆

= 𝑆𝐿𝑈𝑅𝑃𝑆

𝑆𝐿𝑅𝑃

=𝐿𝑈∗

𝑃𝑆∗=𝑆𝑈𝑅𝑆

𝑆𝑐𝑎𝑙𝑒 =𝐿𝑈𝑃𝑆

=412

=4 ÷ 412 ÷ 4

=14

7.△GQA~△EHI𝐺𝐴 = 24 & 𝐸𝐼 = 9.Whatisthescale?

8.△WXC~ △ZTD𝑊𝑋 = 17 & 𝑍𝑇 = 5.Whatisthescale?

9.△MYK~ △JBR𝑀𝐾 = 18 & 𝐽𝑅 = 28.Whatisthescale?

10.△PVF~△NSZ𝑃𝑉 = 3 & 𝑁𝑆 = 6.Whatisthescale?

11.△LAU~△GHS𝐴𝑈 = 8 & 𝐻𝑆 = 4.Whatisthescale?

Sometimes,youareaskedtodetermineifthereisascale(meaning,ifthetrianglesaresimilar).Tofindout,youwouldsubstituteintomorethanonescalefraction.

Ifthesimplifiedfractionsareequal,thenthatisyourscale.IfthesimplifiedfractionsareNOTequal,thenthatisNOTyourscale,becausetheyareNOTSIMILAR.

Determinethescaleforeachfraction,ifthereisone.Ifthescalesarenotequal,thenwrite“NOTSIMILAR.”Example:On△PHO& △NED,𝑃𝐻 = 15,𝐻𝑂 = 18,𝑁𝐸 = 20,𝑃𝑂 = 12,𝐸𝐷 = 18 & 𝑁𝐷 = 16.Whatisthescale?

𝑆𝑐𝑎𝑙𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠:𝑃𝐻𝑂𝑁𝐸𝐷

=𝑃𝐻𝑂𝑁𝐸𝐷

=𝑃𝐻𝑂𝑁𝐸𝐷

𝑆𝑐𝑎𝑙𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠:𝑃𝐻∗

𝑁𝐸∗=𝐻𝑂∗

𝐸𝐷∗=𝑃𝑂∗

𝑁𝐷∗ 𝑆𝑖𝑛𝑐𝑒 𝐼 ℎ𝑎𝑣𝑒 𝑒𝑣𝑒𝑟𝑦 𝑠𝑖𝑑𝑒, 𝐼 𝑤𝑖𝑙𝑙 𝑝𝑙𝑢𝑔 𝑡ℎ𝑒𝑚 𝑎𝑙𝑙 𝑖𝑛:

1520

= 1818

= 1216

𝑁𝑜𝑤, 𝐼 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑒𝑎𝑐ℎ 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑎𝑠 𝑚𝑢𝑐ℎ 𝑎𝑠 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒:15 ÷ 520 ÷ 5

= 18 ÷ 1818 ÷ 18

= 12 ÷ 416 ÷ 4

34

≠ 11

≠ 34

Becausethethreefractionsarenotthesame,thereisnoscale,whichmeansthatthetrianglesare 𝑁𝑂𝑇 𝑆𝐼𝑀𝐼𝐿𝐴𝑅 .

12.On△NMC& △WRK,𝑊𝐾 = 16,𝑁𝑀 = 10,𝑀𝐶 = 5, 𝑁𝐶 = 10,𝑅𝐾 = 8 & 𝑊𝑅 = 16.Whatisthescale?

13.On△BXK& △DZP,𝐵𝑋 = 30,𝐵𝐾 = 36,𝑍𝑃 = 28, 𝑋𝐾 = 24,𝐷𝑍 = 35 & 𝐷𝑃 = 42.Whatisthescale?

14.On△FHS& △LYR,𝐹𝐻 = 22,𝐹𝑆 = 22,𝑌𝑅 = 11, 𝐿𝑌 = 10,𝐻𝑆 = 5 & 𝐿𝑅 = 15.Whatisthescale?

15.On△SVD& △ONS,𝑁𝑆 = 4,𝑂𝑆 = 18,𝑂𝑁 = 16, 𝑆𝑉 = 24,𝑉𝐷 = 21 & 𝑆𝐷 = 27.Whatisthescale?

16.On△AET& △NMW,𝐴𝐸 = 5,𝐸𝑇 = 5,𝑁𝑀 = 4,𝐴𝑇 = 5, 𝑀𝑊 = 4 & 𝑁𝑊 = 4.Whatisthescale?

17.On△BGV& △QCK,𝐵𝐺 = 40,𝑄𝐾 = 5,𝐺𝑉 = 20, 𝐵𝑉 = 50,𝑄𝐶 = 4 & 𝐶𝐾 = 2.Whatisthescale?

18.On△HBQ& △ESZ,𝐸𝑍 = 13, 𝑆𝑍 = 26,𝐸𝑆 = 26, 𝐻𝐵 = 16,𝐵𝑄 = 16,𝐻𝑄 = 8.Whatisthescale?

19.On△FGH& △TVW,𝐺𝐻 = 5,𝐹𝐻 = 1,𝑇𝑊 = 5, 𝑉𝑊 = 15,𝑇𝑉 = 3 & 𝐹𝐺 = 1.Whatisthescale?

20.On△YVM& △WRA,𝑌𝑉 = 10,𝑊𝑅 = 120,𝑉𝑀 = 7, 𝑅𝐴 = 84,𝑌𝑀 = 9 & 𝑊𝐴 = 108.Whatisthescale?