30
SECONDARY MATH I // MODULE 6
CC BY Jorge Jaramillo
6.6 Symmetries of Regular
Polygons
A Solidify Understanding Task
Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedonto
itselfbyarotationissaidtohaverotationalsymmetry.Adiagonalofapolygonisanyline
segmentthatconnectsnon-consecutiveverticesofthepolygon. Foreachofthefollowingregularpolygons,describetherotationsandreflectionsthatcarryitonto
itself:(beasspecificaspossibleinyourdescriptions,suchasspecifyingtheangleofrotation)
1. Anequilateraltriangle
3. Aregularpentagon
4. Aregularhexagon
2. Asquare
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TRANSFORMATIONS AND SYMMETRY – 6.6
31
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
5. Aregularoctagon
6. Aregularnonagon
Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina
regularpolygon?
Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational
symmetryinaregularpolygon?
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
READY, SET, GO!
Name
6.6
PeriodDate
READY
Topic:Rotationalsymmetry,connectedtofractionsofaturnanddegrees.
1.Whatfractionofaturndoesthewagonwheel
belowneedtoturninordertoappearthevery
sameasitdoesrightnow?Howmanydegreesof
rotationwouldthatbe?
2.Whatfractionofaturndoesthepropeller
belowneedtoturninordertoappearthevery
sameasitdoesrightnow?Howmanydegreesof
rotationwouldthatbe?
3.WhatfractionofaturndoesthemodelofaFerriswheelbelowneedtoturninordertoappearthe
verysameasitdoesrightnow?Howmanydegreesof
rotationwouldthatbe?
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SECONDARY MATH I // MODULE 6
6.6
TRANSFORMATIONS AND SYMMETRY – 6.6
SET
Topic:Findinganglesofrotationalsymmetryforregularpolygons,linesofsymmetryanddiagonals
4.Drawthelinesofsymmetryforeachregularpolygon,fillinthetableincludinganexpressionforthe
numberoflinesofsymmetryinan-sidedpolygon.
Numberof
Numberoflines
Sides
ofsymmetry
3
4
5
6
7
8
n
5.Drawallofthediagonalsineachregularpolygon.Fillinthetableandfindapattern,isitlinear,
exponentialorneither?Howdoyouknow?Attempttofindanexpressionforthenumberofdiagonalsin
an-sidedpolygon.
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Numberof
Sides
3
Numberof
diagonals
4
5
6
7
8
n
34
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
6.6
6.Findtheangle(s)ofrotationthatwillcarrythe12sidedpolygonbelowontoitself.
7.Whataretheanglesofrotationfora20-gon?Howmanylinesofsymmetry(linesofreflection)willit
have?
8.Whataretheanglesofrotationfora15-gon?Howmanylineofsymmetry(linesofreflection)willit
have?
9.Howmanysidesdoesaregularpolygonhavethathasanangleofrotationequalto180?Explain.
10.Howmanysidesdoesaregularpolygonhavethathasanangleofrotationequalto200?Howmany
linesofsymmetrywillithave?
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SECONDARY MATH I // MODULE 6
6.6
TRANSFORMATIONS AND SYMMETRY – 6.6
GO
Topic:Reflectingandrotatingpointsonthecoordinateplane.
(Thecoordinategrid,compass,rulerandothertoolsmaybehelpfulindoingthiswork.)
9.ReflectpointAoverthelineofreflection
10.ReflectpointAoverthelineofreflectionand
andlabeltheimageA’.
labeltheimageA’.
line of reflection line of reflection
A A
11.ReflecttriangleABCoverthelineof
12.ReflectparallelogramABCDoverthelineof
reflectionandlabeltheimageA’B’C’.
reflectionandlabeltheimageA’B’C’D’.
B
C
line of reflection
A
D
line of reflection
B
C
A
13.GiventriangleXYZanditsimageX’Y’Z’
14GivenparallelogramQRSTanditsimage
drawthelineofreflectionthatwasused.
Q’R’S’T’drawthelineofreflectionthatwasused.
R
Z
S
R'
S'
Y
T
X
Q
Y'
T'
Q'
X'
Z'
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