In triangle ABC, AD is the angular bisector of the angle A. If AB = 6.4, AC = 8, BD = 5.6, DC = ?.
How can we say that AB/AC = BD/DC?

Asked by neerajmulagapu021 | 11th May, 2015, 03:02: PM

Expert Answer:

 
T o space p r o v e colon space fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator D C end fraction C o n s t r u c t i o n colon space D r a w space C E parallel to D A space t o space m e e t space B A space p r o d u c e d space a t space E.  P r o o f colon S i n c e space C E parallel to D A space a n d space A C space i s space t h e space t r a v e r s a l comma space w e space h a v e angle D A C equals angle A C E space left parenthesis a l t e r n a t e space a n g l e s right parenthesis.... left parenthesis 1 right parenthesis a n d space angle B A D equals angle A E C space left parenthesis c o r r e s p o n d i n g space a n g l e s right parenthesis..... left parenthesis 2 right parenthesis S i n c e space A D space i s space t h e space a n g l e space b i s e c t o r space o f space angle A comma space angle B A D equals angle D A C.... left parenthesis 3 right parenthesis  F r o m space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e space angle A C E equals angle A E C T h u s space i n space triangle A C E comma space w e space h a v e space A E equals A C space left parenthesis s i d e s space o p p o s i t e space t o space e q u a l space a n g l e s space a r e space e q u a l right parenthesis N o w space i n space triangle B C E space w e space h a v e comma space C E parallel to D A rightwards double arrow fraction numerator B D over denominator D C end fraction equals fraction numerator B A over denominator A E end fraction rightwards double arrow fraction numerator B D over denominator D C end fraction equals fraction numerator A B over denominator A C end fraction space left parenthesis because A E equals A C right parenthesis left parenthesis P r o v e d right parenthesis U sin g space t h e space a b o v e space t h e o r e m comma space i n space t h e space g i v e n space t r i a n g l e comma space w e space g e t space fraction numerator B D over denominator D C end fraction equals fraction numerator A B over denominator A C end fraction rightwards double arrow fraction numerator 5.6 over denominator D C end fraction equals fraction numerator 6.4 over denominator 8 end fraction rightwards double arrow D C equals 7

Answered by satyajit samal | 11th May, 2015, 04:43: PM

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