boards concepts booster triangles download doubtnut … · 2018. 12. 8. · 4 concept for boards ||...

BOARDSCONCEPTSBOOSTER

TRIANGLES

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CONCEPTFORBOARDS||ChapterTRIANGLES

1.SIMILARITY

1.Whatwealreadylearnedinpreviousclasses

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1.SIMILARITY

2.Howsimilarityisdifferentfromcongruence.


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1.SIMILARITY

3.SimilarPolygons


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1.SIMILARITY

4.SimilarTrianglesandtheirproperties


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2.SIMILARTRIANGLESANDTHEIRPROPERTIES

1.BasicproportionalityTheoremorThalesTheorem-Ifalineisdrawnparalleltoonesideofatriangleintersectingtheothertwosides;thenitdividesthetwosidesinthesameratio.


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2.If ina ;alineDE||BC;intersectsABinDandACinE;Then / =/


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3. Converse of Basis proportionality theorem : If a line divides any two sides of atriangleinthesameratio;thenthelinemustbeparalleltothethirdside.



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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2. Ifa linethroughonevertexofa triangledividestheoppositesides in theRatioofothertwosides;thenthelinebisectstheangleatthevertex.


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3. The external angle bisector of an angle of a triangle divides the opposite sideexternallyintheratioofthesidescontainingtheangle.


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4.MOREONBASICPROPORTIONALITYTHEOREM

1.Thelinedrawnfromthemidpointofonesideofatriangleparallel toanothersidebisectsthethirdside.


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2.Thelinejoiningthemid-pointsoftwosidesofatriangleisparalleltothethirdside.

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3.Provethatthediagonalsofatrapeziumdivideeachotherproportionally.


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4. If the diagonals of a quadrilateral divide each other proportionally; then it is atrapezium.


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5.Any lineparallel to theparallelsidesofa trapeziumdividesthenon-parallelsidesproportionally.


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6. If three ormore parallel lines are intersected by two transversal; Prove that theinterceptsmadebythemontranversalarepropotional.


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5.CRITERIAFORSIMILARITYOFTRIANGLES

1.AAASimilarityCriterion:Iftwotrianglesareequiangular;thentheyaresimilar


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2.Iftwoanglesofonetrianglearerespectivelyequaltotwoanglesofanothertriangle;thentwotrianglesaresimilar.


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3. SSS Similarity Criterion : If the corresponding sides of two triangles areproportional;thentheyaresimilar


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4. SAS Similarity Criterion : If in two triangle; one pair of corresponding sides areproportionalandtheincludedanglesareequalthentwotrianglesaresimilar.



6.PROPERTIESOFSIMILARTRIANGLE

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21 1.Iftwotrianglesaresimilar;provethattheratioofthecorrespondingsidesissameastheratioofcorrespondingmedians.


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7.MOREONCHARACTERISTICSPROPERTIES

1.Iftwotrianglesaresimilar;provethattheratioofthecorrespondingsidesissameasthecorrespondinganglebisectorsegments.


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2.Iftwotrianglesaresimilar;provethattheratioofcorrespondingsidesisequaltotheratioofcorrespondingaltitudes.




3. Ifoneangleofa triangle isequal tooneangleofanother triangleandbisectorof

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24 theseequalanglesdividetheoppositesideinthesameratio;provethatthetrianglesaresimilar.


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4.Iftwosidesandamedianbisectingoneofthesesidesofatrianglearerespectivelyproportional to the two sides and corresponding median of another triangle; thentrianglearesimilar.


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5. If two sides and a median bisecting the third side of a triangle ar respectivelyproportionaltothecorrespondingsidesandmedianoftheothertriangle;thenthetwotrianglesaresimilar.


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8.AREASOFTWOSIMILARTRIANGLES

1.TheratioofareaofTwosimilartrianglesisequaltotheratioofthesquaresofanytwocorrespondingsides.


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2.Theareaof twosimilar trianglesare in ratioof thesquaresof thecorresponding

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altitudes.


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3. The areas of the two similar triangles are in the ratio of the square of thecorrespondingmedians.


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4.Theareaoftwosimilartriangleareintheratioofthesquareofthecorrespondinganglebisectorsegments


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5.Iftheareaoftwosimilartrianglesareequalthenthetrianglesarecongruent.


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9.PYTHAGORASTHEOREM

1.PYTHAGORASTHEOREM:InaRightangledtriangle;thesquareofhypotenuseisequaltothesumofthesquaresoftheothertwosides.


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9.PYTHAGORASTHEOREM

2. isanobtusetriangle;obtuse-angledatB.If ;provethat


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9.PYTHAGORASTHEOREM

3.Infig. of isanacuteangleand ;provethat


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9.PYTHAGORASTHEOREM

4.Provethatinanytriangle;thesumantthesquaresofanytwosideisequaltotwicethesquareofhalfofthethirdsidetogetherwithtwicethesquareofthemedianwhichbisectsthethirdside.



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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9.PYTHAGORASTHEOREM

6.ConverseofPythagorastheorem:Inatriangle;Ifthesquareofonesideisequaltothesumofthesquaresoftheothertwosides.,thentheangleoppositetothesideisarightangle.


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boards concepts booster triangles download doubtnut … · 2018. 12. 8. · 4 concept for boards ||...

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