What is the orthocenter of a triangle with corners at #(7 ,3 )#, #(4 ,8 )#, and (6 ,8 )#?

Question

What is the orthocenter of a triangle with corners at #(7  ,3  )#, #(4  ,8   )#, and (6   ,8   )#?

Claire Akin · Accepted Answer

The orthocenter is #=(7,42/5)# Let the triangle #DeltaABC# be #A=(7,3)# #B=(4,8)# #C=(6,8)# The slope of the line #BC# is #=(8-8)/(6-4)=0/2=0# The slope of the line perpendicular to #BC# is #=-1/0=-oo# The equation of the line through #A# and perpendicular to #BC# is #x=7#...................#(1)# The slope of the line #AB# is #=(8-3)/(4-7)=5/-2=-5/2# The slope of the line perpendicular to #AB# is #=2/5# The equation of the line through #C# and perpendicular to #AB# is #y-8=2/5(x-6)# #y-8=2/5x-12/5# #y-2/5x=28/5#...................#(2)# Solving for #x# and #y# in equations #(1)# and #(2)# #y-2/5*7=28/5# #y-14/5=28/5# #y=28/5-14/5=42/5# The orthocenter of the triangle is #=(7,42/5)#

Violet Aldrich · Answer

undefined To find the orthocenter of a triangle, we need to determine the point where the three altitudes of the triangle intersect. The altitude of a triangle is a perpendicular line segment from one vertex to the opposite side.

Given the vertices of the triangle as (7, 3), (4, 8), and (6, 8), we first find the slopes of the lines passing through each pair of vertices to determine the equations of the altitudes.

The slopes of the lines passing through the points are:

1. Slope of the line passing through (7, 3) and (4, 8):
   m1 = (8 - 3) / (4 - 7) = 5 / -3 = -5/3

2. Slope of the line passing through (7, 3) and (6, 8):
   m2 = (8 - 3) / (6 - 7) = 5 / -1 = -5

3. Slope of the line passing through (4, 8) and (6, 8):
   m3 = (8 - 8) / (6 - 4) = 0

Now, using the point-slope form of a linear equation, we can find the equations of the altitudes:

1. Equation of the altitude passing through (7, 3):
   y - 3 = (-5/3)(x - 7)

2. Equation of the altitude passing through (4, 8):
   y - 8 = (-5)(x - 4)

3. Equation of the altitude passing through (6, 8):
   y - 8 = 0 (since the line is horizontal)

Next, we find the point of intersection of these lines, which represents the orthocenter of the triangle. Solving the system of equations formed by the three altitude lines will give us the coordinates of the orthocenter.

After solving the system, we find the orthocenter at the point of intersection, which gives the coordinates of the orthocenter of the triangle.