BOARDS CONCEPTS BOOSTER TRIANGLES Download Doubtnut … · 2018. 12. 8. · 4 CONCEPT FOR BOARDS || Chapter TRIANGLES 1. SIMILARITY 4. Similar Triangles and their properties Click

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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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