# What is the orthocenter of a triangle with corners at (2 ,3 ), (5 ,1 ), and (9 ,6 )?

Oct 17, 2016

The Orthocenter is $\left(\frac{121}{23} , \frac{9}{23}\right)$

#### Explanation:

Find the equation of the line that goes through the point $\left(2 , 3\right)$ and is perpendicular to the line through the other two points:

$y - 3 = \frac{9 - 5}{1 - 6} \left(x - 2\right)$

$y - 3 = \frac{4}{- 5} \left(x - 2\right)$

$y - 3 = - \frac{4}{5} x + \frac{8}{5}$

$y = - \frac{4}{5} x + \frac{23}{5}$

Find the equation of the line that goes through the point $\left(9 , 6\right)$ and is perpendicular to the line through the other two points:

$y - 6 = \frac{5 - 2}{3 - 1} \left(x - 9\right)$

$y - 6 = \frac{3}{2} \left(x - 9\right)$

$y - 6 = \frac{3}{2} x - \frac{27}{2}$

$y = \frac{3}{2} x - \frac{15}{2}$

The orthocenter is at the intersection of these two lines:

$y = - \frac{4}{5} x + \frac{23}{5}$
$y = \frac{3}{2} x - \frac{15}{2}$

Because y = y, we set the right sides equal and solve for the x coordinate:

$\frac{3}{2} x - \frac{15}{2} = - \frac{4}{5} x + \frac{23}{5}$

Multiply by 2:

$3 x - 15 = - \frac{8}{5} x + \frac{46}{5}$

Multiply by 5

$15 x - 75 = - 8 x + 46$

$23 x = + 121$

x = 121/23

$y = \frac{3}{2} \left(\frac{121}{23}\right) - \frac{15}{2}$

$y = \frac{3}{2} \left(\frac{121}{23}\right) - \frac{15}{2}$

$y = \frac{363}{46} - \frac{345}{46}$

$y = \frac{9}{23}$

The Orthocenter is $\left(\frac{121}{23} , \frac{9}{23}\right)$