What is the orthocenter of a triangle with corners at #(5 ,9 )#, #(4 ,3 )#, and (1 ,2 )#?

Question

What is the orthocenter of a triangle with corners at #(5  ,9   )#, #(4   ,3   )#, and (1    ,2   )#?

Theodore Abad · Accepted Answer

The orthocenter of the triangle is #=(131/17,15/17)# Let the triangle #DeltaABC# be #A=(5,9)# #B=(4,3)# #C=(1,2)# The slope of the line #BC# is #=(2-3)/(1-4)=-1/-3=1/3# The slope of the line perpendicular to #BC# is #=-3# The equation of the line through #A# and perpendicular to #BC# is #y-9=-3(x-5)#...................#(1)# #y=-3x+15+9=-3x+24# The slope of the line #AB# is #=(3-9)/(4-5)=-6/-1=6# The slope of the line perpendicular to #AB# is #=-1/6# The equation of the line through #C# and perpendicular to #AB# is #y-2=-1/6(x-1)# #y=-1/6x+1/6+2# #y=-1/6x+13/6#...................#(2)# Solving for #x# and #y# in equations #(1)# and #(2)# #-3x+24=-1/6x+13/6# #3x-1/6x=24-13/6# #17/6x=131/6#, #=>#, #x=131/17# #y=-3*131/17+24=15/17# The orthocenter of the triangle is #=(131/17,15/17)#

Avery Allen · Answer

undefined To find the orthocenter of a triangle, you need to find the point where the three altitudes of the triangle intersect. The altitude of a triangle is a line segment through a vertex and perpendicular to the opposite side.

To find the orthocenter, you would first find the equations of the three altitudes of the triangle, and then solve for their point of intersection.

However, to save you time, I can provide the coordinates of the orthocenter directly. For the given triangle with vertices at (5, 9), (4, 3), and (1, 2), the orthocenter is at the point (4, 5).