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

1 Answer
sankarankalyanam
Feb 7, 2018

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

Explanation:

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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)

  1. Once you have the equations from step #1, you can find the slope of the corresponding perpendicular lines.

  2. You will use the slopes you have found from step #2, and the corresponding opposite vertex to find the equations of the 2 lines.

  3. 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)

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