Question
Mon March 30, 2015 By: Divyadutta Behura

If the vertices of a triangle are A(0,4,1), B(2,3,-1) and C(4,5,0), then what is the orthocenter of ABC ?

Expert Reply
Vimala Ramamurthy
Tue March 31, 2015
G i v e n space v e r t i c e s space o f space t h e space t r i a n g l e space A B C space a r e colon A left parenthesis 0 comma 4 comma 1 right parenthesis comma space B open parentheses 2 comma 3 comma negative 1 close parentheses space a n d space C open parentheses 4 comma 5 comma 0 close parentheses L e t space u s space f i n d space A B squared comma space B C squared space a n d space C A squared colon A B squared equals open parentheses 2 minus 0 close parentheses squared plus open parentheses 3 minus 4 close parentheses squared plus open parentheses negative 1 minus 1 close parentheses squared equals 4 plus 1 plus 4 equals 9 B C squared equals open parentheses 2 minus 4 close parentheses squared plus open parentheses 3 minus 5 close parentheses squared plus open parentheses negative 1 minus 0 close parentheses squared equals 4 plus 4 plus 1 equals 9 C A squared equals open parentheses 4 minus 0 close parentheses squared plus open parentheses 5 minus 4 close parentheses squared plus open parentheses 0 minus 1 close parentheses squared equals 16 plus 1 plus 1 equals 18 T h u s comma space w e space h a v e comma space A B squared plus space B C squared equals C A squared rightwards double arrow A B C space i s space a space r i g h t space t r i a n g l e comma space r i g h t space a n g l e d space a t space B F o r space a space r i g h t space t r i a n g l e comma space o r t h o c e n t r e space l i e s space a t space t h e space v e r t e x. S i n c e space t h e space v e r t e x space o f space t h e space r i g h t space t r i a n g l e space l i e s space a t space B comma space B space i s space t h e space o r t h o c e n t r e space o f space t h e space t r i a n g l e. T h u s comma space open parentheses 2 comma 3 comma negative 1 close parentheses space i s space t h e space o r t h o c e n t r e space o f space t h e space t r i a n g l e.
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