Triangle of area same as the area of a quadrilateral

Presentaton showing the construction of a triangle of area equal to the area of a given quadrilateral

1. INNOVATIVEWORK
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2. Doneby
SumeshSS
RegNo—13097
Optional:Mathematics
KUCTEKumarapuram.
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3. AreaofTrianglesand
SomeProperties
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4. ParallelLines
Twolinesonaplanearesaidtobeparalleliftheynever
meettogether.
Thedistancebetweentwoparallellinesissame
everywhere.
C
AB
D
P X
Q Y
PQ=XY
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5. AreaofaRectangle
Theareaofarectangleistheproductofitslengthand
breadth.
AB
C D
Area=AB×BC
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6. AltitudesofaTriangle
Inatrianglethelinedrawnperpendiculartoasidefromthe
vertexoppositetoitiscalledthealtitudetothatside.
Altitudetoallthethreesidescanbedrawn.
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7. A A A
B B BC C C P
Q
R
AP,BQandCRarealtitudestothesidesBC,ACandAB
respectively.
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8. AreaofaTriangle
Areaofanytriangleisgivenby
Area=
1
2
×oneside×altitudetothatside
D
C A
Areaof△ABC=
1
2
×BC×AD
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9. Areausinganyside!
Inatrianglewecanuseanysideanditsaltitudeto×ndthe
area.
Theresultwillbethesame. Sotheareaofthetriangle
△ABCgivenin
page6
canbecalculatedas
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10. Area=
1
2
×BC×AP
or
Area=
1
2
×AC×BQ
or
Area=
1
2
×AB×CR
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11. Generallyinatriangleifwetakeasideasthe
basethenitsareais
Area=
1
2
×base×altitude
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12. AreaGeometrically
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13. Lookatthefollowingtriangle.
Wewillgeometricallyarriveatitsarea!
B C
A
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14. ConsiderBCasthebaseanddrawthealtitude
ADtoBC
B C
D
A
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15. Drawaline
PQ
parallelto
BC
throughthe
midpoint
M
of
AD
B C
D
A
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16. Thiswilldividethetriangleinto3regions–
twotrianglesandatrapezium
B C
D
A
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21. – p. 17/64

22. Thisshowsthat
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23. Thatis
B C
A
D
M
BC
12
AD
Areaoftherectangle=
1
2
×AD×BC
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24. Hencetheareaofthe
giventriangle = areaoftheobtainedrectangle.
=
1
2
×AD×BC
Areaofatriangle=
1
2
×oneside×Altitudetotheside
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25. TrianglesOfEqualArea
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26. Considerthefollowingtriangles LM
XY
AB
C
PQ
R Here
XYkLMandAB=PQ
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27. Canyouseeanyrelationbetweenthe
triangles?
Now!lookatthefollowinggures.
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28. LM
XY
AB
C
PQ
R
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29. LM
XY
AB
C
PQ
R
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30. LM
XY
AB
C
PQ
R
D S
HereAB=PQandCD=RS(sincethelinesLMandXY
areparallel). Sotheareasofthetrianglesarethesame!
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31. Nowlookatthefollowingtriangles
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32. LM
XY
A B
PQ
InthesetwotrianglesthesideABiscommon. Consider
thisasthecommonbaseforthetriangles. Nowmove
forward
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33. LM
XY
A B
PQ
LM
XY
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34. LM
XY
A B
PQ
LM
XY
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35. LM
XY
A B
PQ
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36. LM
XY
A B
PQ
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37. LM
XY
A B
PQ
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38. LM
XY
A B
P
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39. LM
XY
A B
P
Sotheareaofthetrianglesareequal.
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40. Thesetwotriangleshavethesamebaseandare
standbetweenapairofparallellinesandthey
havesamearea.
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41. “Iftwotriangleshaveacommonbase
andstandbetweenthesamepairof
parallellines,thentheirareasareequal"
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42. Canyouconstructmoretriangles
ofareaequaltotheareaofthe
giventriangle?
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43. Constructthreetrianglesofareaequaltothe
areaofthefollowingtriangle?
Trytodo.Ifyoufeeldifcultygotothenextpage.
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49. Inthegurelengthoftherectangleis10cmanditsbreadth
is7cmlong. Whatistheareaoftheyellowcoloured
portion?
A B
C D
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50. Whatistherelationbetweentheareasof
△ABCandtheyellowcolouredportion?
Tryagain!
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51. TriangleswithAreaEqualtothe
AreaofGivenQuadrilateral
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52. Considerthefollowingquadrilateral.
A B
C
D
Weshallmakeatriangleofequalareatotheabove
quadrilateral.
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53. A B
C
D
AreaofthequadrilateralABCD= Areaof∆ABD+
Areaof∆BCD
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54. A
B
C
D
NowdrawingalineparalleltoBD.
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55. A
B
C
D
P
AreaofthequadrilateralABCD= Areaof∆ABD+
Areaof∆BCD
= Areaof∆ABD+
Areaof∆BPD
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56. A
B
C
D
P
AreaofthequadrilateralABCD= Areaof∆ABD+
Areaof∆BPD
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57. A
B
C
D
P
AreaofthequadrilateralABCD= Areaof∆ABD+
Areaof∆BPD
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58. A
B
C
D
P
AreaofthequadrilateralABCD= Areaof∆ABD+
Areaof∆BPD
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59. A
B
C
D
P
AreaofthequadrilateralABCD=Areaof∆APD
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60. A
D
P
AreaofthequadrilateralABCD=Areaof∆APD
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61. Howwillyouconstructatriangleofareaequaltothegiven
quadrilateral?
Lookatthenextpage.
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62. Firstconstructthegivenquadrilateral.
A
B
C
D
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63. ThenjointhediagonalBD
A
B
C
D
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64. NowdrawalineparalleltoBDthroughC.
A
B
C
D
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65. ExtentthesideABtowardsthepointE
A
B
C
D
E
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66. NowjoinDE
A
B
C
D
E
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67. Thuswegot∆ADEofareaequaltoareaofthe
quadrilateralABCD
A
D
E
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68. EndofthePresentation
ThankYou!
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