The vertices of the triangle are A(5,4,6), B(1,-1,3) and C(4,3,2). The internal bisector of angle A meet BC at D. Find the co-ordinates of D and the length AD.

Asked by shaktidevgun | 12th Dec, 2015, 06:52: PM

Expert Answer:

L e t space triangle A B C space b e space a space t r i a n g l e space a n d 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. A n g l e space b i s e c t o r space t h e o r e m space s t a t e s space t h a t comma A n space a n g l e space b i s e c t o r space o f space a n space a n g l e space o f space a space t r i a n g l e space d i v i d e s space t h e space o p p o s i t e space s i d e space i n space t w o space s e g m e n t s space t h a t space a r e space p r o p o r t i o n a l space t o space t h e space o t h e r space t w o space s i d e s space o f space t h e space t r i a n g l e. O b s e r v e space t h e space f o l l o w i n g space f i g u r e.
T h u s comma space b y space a n g l e space b i s e c t o r space t h e o r e m comma space w e space h a v e comma fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator C D end fraction  A left parenthesis 5 comma 4 comma 6 right parenthesis comma space B left parenthesis 1 comma negative 1 comma 3 right parenthesis space a n d space C left parenthesis 4 comma 3 comma 2 right parenthesis A B equals square root of open parentheses 1 minus 5 close parentheses squared plus open parentheses negative 1 minus 4 close parentheses squared plus open parentheses 3 minus 6 close parentheses squared end root equals square root of 16 plus 25 plus 9 end root equals 5 square root of 2 space u n i t s A C equals square root of open parentheses 4 minus 5 close parentheses squared plus open parentheses 3 minus 4 close parentheses squared plus open parentheses 2 minus 6 close parentheses squared end root equals square root of 1 plus 1 plus 16 end root equals 3 square root of 2 space u n i t s T h u s comma space fraction numerator A B over denominator A C end fraction equals fraction numerator B D over denominator C D end fraction equals fraction numerator 5 square root of 2 over denominator 3 square root of 2 end fraction rightwards double arrow fraction numerator B D over denominator C D end fraction equals 5 over 3 T h u s comma space t h e space p o i n t space D space d i v i d e s space t h e space l i n e space s e g m e n t space j o i n i n g space t h e space p o i n t s space B left parenthesis 1 comma negative 1 comma 3 right parenthesis space a n d space C left parenthesis 4 comma 3 comma 2 right parenthesis space i n space t h e space r a t i o space 5 colon 3 L e t space t h e space c o o r d i n a t e s space o f space D space b e space D open parentheses x comma y comma z close parentheses T h u s space b y space u sin g space s e c t i o n space f o r m u l a comma space w e space h a v e x equals fraction numerator m x subscript 2 plus n x subscript 1 over denominator m plus n end fraction equals fraction numerator 5 cross times 4 plus 3 cross times 1 over denominator 5 plus 3 end fraction equals 23 over 8 y equals fraction numerator m y subscript 2 plus n y subscript 1 over denominator m plus n end fraction equals fraction numerator 5 cross times 3 minus 3 cross times 1 over denominator 5 plus 3 end fraction equals 12 over 8 y equals fraction numerator m z subscript 2 plus n z subscript 1 over denominator m plus n end fraction equals fraction numerator 5 cross times 2 plus 3 cross times 3 over denominator 5 plus 3 end fraction equals 19 over 8  T h u s space t h e space c o o r d i n a t e s space o f space D space a r e space D open parentheses 23 over 8 comma 12 over 8 comma 19 over 8 close parentheses G i v e n space c o o r d i n a t e s space o f space A space a r e space A left parenthesis 5 comma 4 comma 6 right parenthesis W e space n e e d space t o space f i n d space t h e space l e n g t h space o f space A D. A D equals square root of open parentheses 5 minus 23 over 8 close parentheses squared plus open parentheses 4 minus 12 over 8 close parentheses squared plus open parentheses 6 minus 19 over 8 close parentheses squared end root space space space space equals square root of open parentheses fraction numerator 40 minus 23 over denominator 8 end fraction close parentheses squared plus open parentheses fraction numerator 32 minus 12 over denominator 8 end fraction close parentheses squared plus open parentheses fraction numerator 48 minus 19 over denominator 8 end fraction close parentheses squared end root space space space space equals square root of open parentheses 17 over 8 close parentheses squared plus open parentheses 20 over 8 close parentheses squared plus open parentheses 29 over 8 close parentheses squared end root space space space space equals 1 over 8 square root of 289 plus 400 plus 841 end root space space space space equals 1 over 8 square root of 1530 T h u s comma space A D equals 1 over 8 square root of 1530 space u n i t s
 
 

Answered by Vimala Ramamurthy | 13th Dec, 2015, 11:04: PM