# What is the orthocenter of a triangle with corners at (9 ,3 ), (6 ,9 ), and (2 ,4 )?

Aug 3, 2018

color(maroon)("ortho-centre coordinates " O (73/13, 82/13)#

#### Explanation:

$A \left(9 , 3\right) , B \left(6 , 9\right) , C \left(2 , 4\right)$

Slope of $\overline{A B} = {m}_{A B} = \frac{{y}_{B} - {y}_{A}}{{x}_{B} - {x}_{A}} = \frac{9 - 3}{6 - 9} = - 2$

Slope of $\overline{C F} = {m}_{C F} = - \frac{1}{m} \left(A B\right) = - \frac{1}{-} 2 = \frac{1}{2}$

Equation of $\overline{C F}$ is $y - 4 = \frac{1}{2} \left(x - 2\right)$

$2 y - x = 7$ Eqn (1)

Slope of $\overline{A C} = {m}_{A C} = \frac{{y}_{C} - {y}_{A}}{{x}_{C} - {x}_{A}} = \frac{4 - 3}{2 - 9} = - \frac{1}{7}$

Slope of $\overline{B E} = {m}_{B E} = - \frac{1}{m} \left(A C\right) = - \frac{1}{- \frac{1}{7}} = 7$

Equation of $\overline{B E}$ is $y - 9 = 7 \left(x - 6\right)$

$7 x - y = 33$ Eqn (2)

Solving Eqns (1) and (2), we get the ortho-centre coordinates $O \left(x , y\right)$

$\cancel{2 y} - x + 14 x - \cancel{2 y} = 7 + 66$

$x = \frac{73}{13}$

$y = \frac{164}{26} = \frac{82}{13}$