What is the orthocenter of a triangle with corners at #(9 ,7 )#, #(4 ,1 )#, and (8 ,2 )#?

Question

What is the orthocenter of a triangle with corners at #(9  ,7   )#, #(4   ,1   )#, and (8    ,2   )#?

Savannah Aguilar · Accepted Answer

The orthocenter of the triangle is #=(206/19,-7/19)# Let the triangle #DeltaABC# be #A=(9,7)# #B=(4,1)# #C=(8,2)# The slope of the line #BC# is #=(2-1)/(8-4)=1/4# The slope of the line perpendicular to #BC# is #=-4# The equation of the line through #A# and perpendicular to #BC# is #y-7=-4(x-9)#...................#(1)# #y=-4x+36+7=-4x+43# The slope of the line #AB# is #=(1-7)/(4-9)=-6/-5=6/5# The slope of the line perpendicular to #AB# is #=-5/6# The equation of the line through #C# and perpendicular to #AB# is #y-2=-5/6(x-8)# #y-2=-5/6x+20/3# #y+5/6x=20/3+2=26/3#...................#(2)# Solving for #x# and #y# in equations #(1)# and #(2)# #-4x+43=26/3-5/6x# #4x-5/6x=43-26/3# #19/6x=103/3# #x=206/19# #y=26/3-5/6x=26/3-5/6*206/19=26/3-1030/114=-42/114=-7/19# The orthocenter of the triangle is #=(206/19,-7/19)#

Allison Allen · Answer

undefined To find the orthocenter of a triangle, follow these steps:

1. Calculate the slopes of the lines passing through each pair of vertices.
2. Determine the equations of the perpendicular bisectors of each side of the triangle.
3. Find the point of intersection of any two perpendicular bisectors. This point is the circumcenter of the triangle.
4. Calculate the slopes of the lines passing through each vertex and the circumcenter.
5. Use the point-slope form to find the equations of the altitudes passing through each vertex.
6. Find the point of intersection of any two altitudes. This point is the orthocenter of the triangle.

Applying these steps to the given triangle with vertices (9, 7), (4, 1), and (8, 2), you can find the orthocenter.