What is the orthocenter of a triangle with corners at #(4 ,1 )#, #(7 ,4 )#, and (2 ,8 )#?

1 Answer
Jan 22, 2018

#(53/18, 71/18)#

Explanation:

1) Find the slope of two lines.
#(4,1) and (7,4)#
#m_1 = 1#
#(7,4) and (2,8)#
#m_2 = -4/5#

2) Find the perpendicular of both slopes.
#m_(perp1) = -1#
#m_(perp2) = 5/4#

3) Find the midpoints of the points you used.
#(4,1) and (7,4)#
#mid_1# = #(11/2,3/2)#
#(7,4) and (2,8)#
#mid_2# = #(9/2,6)#

4) Using the slope, find an equation that fits it.
#m=-1#, point = #(11/2, 3/2)#
#y=-x+b#
#3/2=-11/2+b#
#b=7#

#y=-x+7# #=> 1#

#m=5/4#, point = #(9/2,6)#
#y=5/4x+b#
#6=9/2*5/4+b#
#6=45/8+b#
#b=3/8#

#y=5/4x+3/8# #=> 2#

4) Set does equations equal to each other.
#-x+7 = 5/4x+3/8#
#9/4x = 53/8#
#18x=53#
#x=53/18#

5) Plug in the x-value and solve for y
#y=-x+7#
#y=-53/18+7#
#y=73/18#

6) The answer is...
#(53/18, 71/18)#