# What is the orthocenter of a triangle with corners at (6 ,3 ), (2 ,4 ), and (7 ,9 )#?

Jan 20, 2018

Orthocenter of the triangle is at $\left(5.6 , 3.4\right)$

#### Explanation:

Orthocenter is the point where the three "altitudes" of a triangle meet. An "altitude" is a line that goes through a vertex (corner point) and is at right angles to the opposite side.

$A = \left(6 , 3\right) , B \left(2 , 4\right) , C \left(7 , 9\right)$ . Let $A D$ be the altitude from $A$ on $B C$ and $C F$ be the altitude from $C$ on $A B$ they meet at point $O$ , the orthocenter.

Slope of $B C$ is ${m}_{1} = \frac{9 - 4}{7 - 2} = \frac{5}{5} = 1$

Slope of perpendicular $A D$ is ${m}_{2} = - 1 \left({m}_{1} \cdot {m}_{2} = - 1\right)$

Equation of line $A D$ passing through $A \left(6 , 3\right)$ is

$y - 3 = - 1 \left(x - 6\right) \mathmr{and} y - 3 = - x + 6 \mathmr{and} x + y = 9 \left(1\right)$

Slope of $A B$ is ${m}_{1} = \frac{4 - 3}{2 - 6} = - \frac{1}{4}$

Slope of perpendicular $C F$ is ${m}_{2} = - \frac{1}{- \frac{1}{4}} = 4$

Equation of line $C F$ passing through $C \left(7 , 9\right)$ is

$y - 9 = 4 \left(x - 7\right) \mathmr{and} y - 9 = 4 x - 28 \mathmr{and} 4 x - y = 19 \left(2\right)$

Solving equation(1) and (2) we get their intersection point , which

is the orthocenter. Adding equation(1) and (2) we get,

$5 x = 28 \mathmr{and} x = \frac{28}{5} = 5.6 \mathmr{and} y = 9 - x = 9 - 5.6 = 3.4$

Orthocenter of the triangle is at $\left(5.6 , 3.4\right)$ [Ans]