# What is the orthocenter of a triangle with corners at (6 ,3 ), (4 ,5 ), and (2 ,9 )?

Jul 2, 2018

The orthocenter of triangle is $\left(- 14 , - 7\right)$

#### Explanation:

Let $\triangle A B C$ be the triangle with corners at

$A \left(6 , 3\right) , B \left(4 , 5\right) \mathmr{and} C \left(2 , 9\right)$

Let $\overline{A L} , \overline{B M} \mathmr{and} \overline{C N}$ be the altitudes of sides

$\overline{B C} , \overline{A C} , \mathmr{and} \overline{A B}$ respectively.

Let $\left(x , y\right)$ be the intersection of three altitudes .

Slope of $\overline{A B} = \frac{5 - 3}{4 - 6} = - 1$

$\overline{A B} \bot \overline{C N} \implies$slope of $\overline{C N} = 1$ , $\overline{C N}$ passes through $C \left(2 , 9\right)$

$\therefore$The equn. of $\overline{C N}$ is $: y - 9 = 1 \left(x - 2\right)$

i.e. color(red)(x-y=-7.....to (1)

Slope of $\overline{B C} = \frac{9 - 5}{2 - 4} = - 2$

$\overline{A L} \bot \overline{B C} \implies$slope of $\overline{A L} = \frac{1}{2}$ , $\overline{A L}$ passes through $A \left(6 , 3\right)$

$\therefore$The equn. of $\overline{A L}$ is $: y - 3 = \frac{1}{2} \left(x - 6\right) \implies 2 y - 6 = x - 6$

i.e. color(red)(x=2y.....to (2)

Subst. $x = 2 y$ into $\left(1\right)$ ,we get

2y-y=-7=>color(blue)( y=-7

From equn.$\left(2\right)$ we get

x=2y=2(-7)=>color(blue)(x=-14#

Hence, the orthocenter of triangle is $\left(- 14 , - 7\right)$