What is the orthocenter of a triangle with corners at #(5 ,4 )#, #(2 ,3 )#, and (7 ,8 )#?

Question

What is the orthocenter of a triangle with corners at #(5  ,4   )#, #(2   ,3   )#, and (7    ,8   )#?

Violet Aldrich · Accepted Answer

The orthocenter is #=(10,-1)# Let the triangle #DeltaABC# be #A=(5,4)# #B=(2,3)# #C=(7,8)# The slope of the line #BC# is #=(8-3)/(7-2)=5/5=1# The slope of the line perpendicular to #BC# is #=-1# The equation of the line through #A# and perpendicular to #BC# is #y-4=-1(x-5)# #y-4=-x+5# #y+x=9#...................#(1)# The slope of the line #AB# is #=(3-4)/(2-5)=-1/-3=1/3# The slope of the line perpendicular to #AB# is #=-3# The equation of the line through #C# and perpendicular to #AB# is #y-8=-3(x-7)# #y-8=-3x+21# #y+3x=29#...................#(2)# Solving for #x# and #y# in equations #(1)# and #(2)# #y+3(9-y)=29# #y+27-3y=29# #-2y=29-27=2# #y=-2/2=-1# #x=9-y=9+1=10# The orthocenter of the triangle is #=(10,-1)#

Aurora Ayala · Answer

undefined To find the orthocenter of a triangle, we need to first determine the altitudes of the triangle. The orthocenter is the point where the three altitudes intersect.

Let's label the vertices of the triangle as $ A(5, 4) $, $ B(2, 3) $, and $ C(7, 8) $.

1. Find the slopes of the line segments formed by pairs of vertices to get the slopes of the sides of the triangle.
2. Determine the perpendicular slopes to these sides to find the equations of the altitudes passing through each vertex.
3. Find the point of intersection of these altitudes, which is the orthocenter.

Let's proceed with the calculations:

1. Slopes of the sides:

- Slope of $ AB $:
  $$ m_{AB} = \frac{3 - 4}{2 - 5} = \frac{-1}{-3} = \frac{1}{3} $$
- Slope of $ BC $:
  $$ m_{BC} = \frac{8 - 3}{7 - 2} = \frac{5}{5} = 1 $$
- Slope of $ AC $:
  $$ m_{AC} = \frac{8 - 4}{7 - 5} = \frac{4}{2} = 2 $$

2. Perpendicular slopes (negative reciprocals):

- Perpendicular slope of $ AB $: $ -\frac{1}{3} $
- Perpendicular slope of $ BC $: $ -1 $
- Perpendicular slope of $ AC $: $ -\frac{1}{2} $

3. Equations of altitudes:

- Altitude from $ A $: $ y - 4 = -\frac{1}{3}(x - 5) $
- Altitude from $ B $: $ y - 3 = -1(x - 2) $
- Altitude from $ C $: $ y - 8 = -\frac{1}{2}(x - 7) $

Now, find the point of intersection of these lines to get the orthocenter. Once you have the coordinates of the orthocenter, you'll have the solution.