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Triangle ABC has the vertices (1,2), (3,5), and (6,3)  Find the coordinates of the orthocenter.

 Feb 19, 2021
 #1
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We  only  need  the equations  of two lines  to do this

Find  the slope   between  AB  =   (1,2)  and (3,5)  =  (5-2)  /( 3 -1)  =  3/2

The equation  of  the line perpendicular  to segment  AB  through  ( 6,3)  is

 

y  = (-2/3)(x - 6) + 3

y = (-2/3)x + 4  +  3

y = (-2/3)x  + 7        (1)

 

Next

Find  the slope  between  AC  = ( 1,2)  and (6,3)  =  (3 -2) /( 6 - 1)  =  1/5

The equation of  the  line  perpendicular  to segment AC  through  B  is

 

y = -5 ( x -3)+  5

y = -5x + 20        (2)

 

Find  the  x coordinate  of the  orthocenter  by  setting  (1)   = (2)

 

(-2/3)x  +  7   =  - 5x  +  20     multiply through  by  3

 

-2x  + 21  = -15x  +  60

 

13x  =  39

 

x  = 3

 

And  the y coordinate  is   -5(3)  +  20  =   5

 

So....the orthocenter is  actually B    = (3,5)

 

 

cool cool cool

 Feb 19, 2021

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