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BOARDSCONCEPTSBOOSTER
TRIANGLES
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1
CONCEPTFORBOARDS||ChapterTRIANGLES
1.SIMILARITY
1.Whatwealreadylearnedinpreviousclasses
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CONCEPTFORBOARDS||ChapterTRIANGLES
1.SIMILARITY
2.Howsimilarityisdifferentfromcongruence.
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CONCEPTFORBOARDS||ChapterTRIANGLES
1.SIMILARITY
3.SimilarPolygons
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CONCEPTFORBOARDS||ChapterTRIANGLES
1.SIMILARITY
4.SimilarTrianglesandtheirproperties
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CONCEPTFORBOARDS||ChapterTRIANGLES
2.SIMILARTRIANGLESANDTHEIRPROPERTIES
1.BasicproportionalityTheoremorThalesTheorem-Ifalineisdrawnparalleltoonesideofatriangleintersectingtheothertwosides;thenitdividesthetwosidesinthesameratio.
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CONCEPTFORBOARDS||ChapterTRIANGLES
2.SIMILARTRIANGLESANDTHEIRPROPERTIES
2.If ina ;alineDE||BC;intersectsABinDandACinE;Then / =/
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CONCEPTFORBOARDS||ChapterTRIANGLES
2.SIMILARTRIANGLESANDTHEIRPROPERTIES
3. Converse of Basis proportionality theorem : If a line divides any two sides of atriangleinthesameratio;thenthelinemustbeparalleltothethirdside.
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CONCEPTFORBOARDS||ChapterTRIANGLES
ΔABC AB ADAC AE
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3.INTERNALANDEXTERNALBISECTORSOFANANGLEOFATRIANGLE
1. The internal angle bisector of an angle of a triangle divide the opposite sideinternallyintheratioofthesidescontaingtheangle
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CONCEPTFORBOARDS||ChapterTRIANGLES
3.INTERNALANDEXTERNALBISECTORSOFANANGLEOFATRIANGLE
2. Ifa linethroughonevertexofa triangledividestheoppositesides in theRatioofothertwosides;thenthelinebisectstheangleatthevertex.
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CONCEPTFORBOARDS||ChapterTRIANGLES
3.INTERNALANDEXTERNALBISECTORSOFANANGLEOFATRIANGLE
3. The external angle bisector of an angle of a triangle divides the opposite sideexternallyintheratioofthesidescontainingtheangle.
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CONCEPTFORBOARDS||ChapterTRIANGLES
4.MOREONBASICPROPORTIONALITYTHEOREM
1.Thelinedrawnfromthemidpointofonesideofatriangleparallel toanothersidebisectsthethirdside.
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CONCEPTFORBOARDS||ChapterTRIANGLES
4.MOREONBASICPROPORTIONALITYTHEOREM
2.Thelinejoiningthemid-pointsoftwosidesofatriangleisparalleltothethirdside.
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CONCEPTFORBOARDS||ChapterTRIANGLES
4.MOREONBASICPROPORTIONALITYTHEOREM
3.Provethatthediagonalsofatrapeziumdivideeachotherproportionally.
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CONCEPTFORBOARDS||ChapterTRIANGLES
4.MOREONBASICPROPORTIONALITYTHEOREM
4. If the diagonals of a quadrilateral divide each other proportionally; then it is atrapezium.
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CONCEPTFORBOARDS||ChapterTRIANGLES
4.MOREONBASICPROPORTIONALITYTHEOREM
5.Any lineparallel to theparallelsidesofa trapeziumdividesthenon-parallelsidesproportionally.
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CONCEPTFORBOARDS||ChapterTRIANGLES
4.MOREONBASICPROPORTIONALITYTHEOREM
6. If three ormore parallel lines are intersected by two transversal; Prove that theinterceptsmadebythemontranversalarepropotional.
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CONCEPTFORBOARDS||ChapterTRIANGLES
5.CRITERIAFORSIMILARITYOFTRIANGLES
1.AAASimilarityCriterion:Iftwotrianglesareequiangular;thentheyaresimilar
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CONCEPTFORBOARDS||ChapterTRIANGLES
5.CRITERIAFORSIMILARITYOFTRIANGLES
2.Iftwoanglesofonetrianglearerespectivelyequaltotwoanglesofanothertriangle;thentwotrianglesaresimilar.
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CONCEPTFORBOARDS||ChapterTRIANGLES
5.CRITERIAFORSIMILARITYOFTRIANGLES
3. SSS Similarity Criterion : If the corresponding sides of two triangles areproportional;thentheyaresimilar
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CONCEPTFORBOARDS||ChapterTRIANGLES
5.CRITERIAFORSIMILARITYOFTRIANGLES
4. SAS Similarity Criterion : If in two triangle; one pair of corresponding sides areproportionalandtheincludedanglesareequalthentwotrianglesaresimilar.
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CONCEPTFORBOARDS||ChapterTRIANGLES
6.PROPERTIESOFSIMILARTRIANGLE
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21 1.Iftwotrianglesaresimilar;provethattheratioofthecorrespondingsidesissameastheratioofcorrespondingmedians.
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CONCEPTFORBOARDS||ChapterTRIANGLES
7.MOREONCHARACTERISTICSPROPERTIES
1.Iftwotrianglesaresimilar;provethattheratioofthecorrespondingsidesissameasthecorrespondinganglebisectorsegments.
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CONCEPTFORBOARDS||ChapterTRIANGLES
7.MOREONCHARACTERISTICSPROPERTIES
2.Iftwotrianglesaresimilar;provethattheratioofcorrespondingsidesisequaltotheratioofcorrespondingaltitudes.
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CONCEPTFORBOARDS||ChapterTRIANGLES
7.MOREONCHARACTERISTICSPROPERTIES
3. Ifoneangleofa triangle isequal tooneangleofanother triangleandbisectorof
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24 theseequalanglesdividetheoppositesideinthesameratio;provethatthetrianglesaresimilar.
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CONCEPTFORBOARDS||ChapterTRIANGLES
7.MOREONCHARACTERISTICSPROPERTIES
4.Iftwosidesandamedianbisectingoneofthesesidesofatrianglearerespectivelyproportional to the two sides and corresponding median of another triangle; thentrianglearesimilar.
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CONCEPTFORBOARDS||ChapterTRIANGLES
7.MOREONCHARACTERISTICSPROPERTIES
5. If two sides and a median bisecting the third side of a triangle ar respectivelyproportionaltothecorrespondingsidesandmedianoftheothertriangle;thenthetwotrianglesaresimilar.
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CONCEPTFORBOARDS||ChapterTRIANGLES
8.AREASOFTWOSIMILARTRIANGLES
1.TheratioofareaofTwosimilartrianglesisequaltotheratioofthesquaresofanytwocorrespondingsides.
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CONCEPTFORBOARDS||ChapterTRIANGLES
8.AREASOFTWOSIMILARTRIANGLES
2.Theareaof twosimilar trianglesare in ratioof thesquaresof thecorresponding
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altitudes.
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CONCEPTFORBOARDS||ChapterTRIANGLES
8.AREASOFTWOSIMILARTRIANGLES
3. The areas of the two similar triangles are in the ratio of the square of thecorrespondingmedians.
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CONCEPTFORBOARDS||ChapterTRIANGLES
8.AREASOFTWOSIMILARTRIANGLES
4.Theareaoftwosimilartriangleareintheratioofthesquareofthecorrespondinganglebisectorsegments
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CONCEPTFORBOARDS||ChapterTRIANGLES
8.AREASOFTWOSIMILARTRIANGLES
5.Iftheareaoftwosimilartrianglesareequalthenthetrianglesarecongruent.
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CONCEPTFORBOARDS||ChapterTRIANGLES
9.PYTHAGORASTHEOREM
1.PYTHAGORASTHEOREM:InaRightangledtriangle;thesquareofhypotenuseisequaltothesumofthesquaresoftheothertwosides.
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CONCEPTFORBOARDS||ChapterTRIANGLES
9.PYTHAGORASTHEOREM
2. isanobtusetriangle;obtuse-angledatB.If ;provethat
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CONCEPTFORBOARDS||ChapterTRIANGLES
9.PYTHAGORASTHEOREM
3.Infig. of isanacuteangleand ;provethat
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CONCEPTFORBOARDS||ChapterTRIANGLES
9.PYTHAGORASTHEOREM
4.Provethatinanytriangle;thesumantthesquaresofanytwosideisequaltotwicethesquareofhalfofthethirdsidetogetherwithtwicethesquareofthemedianwhichbisectsthethirdside.
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CONCEPTFORBOARDS||ChapterTRIANGLES
9.PYTHAGORASTHEOREM
△ ABC AD ⊥ CB(AC)2 = (AB)2
+ (BC)2 + 2BC. BD
∠B △ ABC AD ⊥ BCAC 2 = AB2 + BC2
− 2BC × BD
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36 5.Threetimesthesumofsquareofthesidesofatriangleisequaltofourtimesthesumofthesquareofthemediansofthetriangle.
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CONCEPTFORBOARDS||ChapterTRIANGLES
9.PYTHAGORASTHEOREM
6.ConverseofPythagorastheorem:Inatriangle;Ifthesquareofonesideisequaltothesumofthesquaresoftheothertwosides.,thentheangleoppositetothesideisarightangle.
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