What is the orthocenter of a triangle with corners at #(3 ,1 )#, #(1 ,6 )#, and (5 ,2 )#?

Given:

Vertices at #(3 ,1 )#, #(1 ,6 )#, and #(5 ,2 )#.

We have three vertices: #color(blue)(A(3,1), B(1,6) and C(5,2)#.

#color(green)(ul (Step:1#

We will find the slope using the vertices #A(3,1), and B(1,6)#.

Let #(x_1, y_1) = (3,1) and (x_2, y_2) = (1,6)#

Formula to find the slope (m) = #color(red)((y_2-y_1)/(x_2-x_1)#

#m=(6-1)/(1-3)#

#m=-5/2#

We need a perpendicular line from the vertex #C# to intersect with the side #AB# at #90^@# angle. To do that, we must find the perpendicular slope, which is the opposite reciprocal of our slope #(m)=-5/2#.

Perpendicular slope is #=-(-2/5) = 2/5#

#color(green)(ul (Step:2#

Use the Point-Slope Formula to find the equation.

Point-slope formula: #color(blue)(y=m(x-h)+k#, where
#m# is the perpendicular slope and #(h,k)# represent the vertex #C# at #(5, 2)#

Hence, #y=(2/5)(x-5)+2#

#y=2/5x-10/5+2#

#y=2/5x# #" "color(red)(Equation.1#

#color(green)(ul (Step:3#

We will repeat the process from #color(green)(ul (Step:1# and #color(green)(ul (Step:2#

Consider side #AC#. Vertices are #A(3,1) and C(5,2)#

Next, we find the slope.

#m=(2-1)/(5-3)#

#m=1/2#

Find the perpendicular slope.

#=rArr -(2/1) =- 2#

#color(green)(ul (Step:4#

Point-slope formula: #color(blue)(y=m(x-h)+k#, using the vertex #B# at #(1, 6)#

Hence, #y=(-2)(x-1)+6#

#y= -2x+8# #" "color(red)(Equation.2#

#color(green)(ul (Step:5#

Find the solution to the system of linear equations to find the vertices of the Orthocenter of the triangle.

#y=2/5x# #" "color(red)(Equation.1#

#y= -2x+8# #" "color(red)(Equation.2#

The solution is becoming too long. Method of Substitution will provide solution for the system of linear equations.

Orthocenter #=(10/3, 4/3)#

The construction of the triangle with the Orthocenter is: