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

Question

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

Claire Akin · Accepted Answer

Orthocenter of the triangle ABC is #color(green)(H (14/5, 9/5)#

The steps to find the orthocenter are:
1. Find the equations of 2 segments of the triangle (for our example we will find the equations for AB, and BC)

Once you have the equations from step #1, you can find the slope of the corresponding perpendicular lines.
You will use the slopes you have found from step #2, and the corresponding opposite vertex to find the equations of the 2 lines.
Once you have the equation of the 2 lines from step #3, you can solve the corresponding x and y, which is the coordinates of the orthocenter.

Given (A(3,1), B(4,5), C(2,2)

Slope of AB #m_c = (y_B - y_A) / (x_B - x_A) = (5-1) / (4-3) = 4#

Slope of #AH_C# #m_(CH_C) = -1 / m_(AB) = -1/4#

Similarly, slope of BC #m_a = (2-4) / (2-5) = 2/3#

Slope of #(AH_A)# #m_(AH_A) = (-1 / (2/3) = -3/2#

Equation of #CH_C#

#y - 2 = -(1/4) (x - 2)#

#4y + x = 10# eqn (1)

Equation of #AH_A#

#y - 1 =- (3/2) (x - 3)#

#2y + 3x = 12# Eqn (1)

Solving equations (1), (2), we get the coordinates of Orthocenter H.

#color(green)(H (14/5, 9/5)#