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

Mar 17, 2018

Triangle with vertices at $\left(3 , 1\right)$, $\left(1 , 6\right)$, and $\left(5 , 2\right)$.

Orthocenter = color(blue)((3.33, 1.33)

#### Explanation:

Given:

Vertices at $\left(3 , 1\right)$, $\left(1 , 6\right)$, and $\left(5 , 2\right)$.

We have three vertices: color(blue)(A(3,1), B(1,6) and C(5,2).

color(green)(ul (Step:1

We will find the slope using the vertices $A \left(3 , 1\right) , \mathmr{and} B \left(1 , 6\right)$.

Let $\left({x}_{1} , {y}_{1}\right) = \left(3 , 1\right) \mathmr{and} \left({x}_{2} , {y}_{2}\right) = \left(1 , 6\right)$

Formula to find the slope (m) = color(red)((y_2-y_1)/(x_2-x_1)

$m = \frac{6 - 1}{1 - 3}$

$m = - \frac{5}{2}$

We need a perpendicular line from the vertex $C$ to intersect with the side $A B$ at ${90}^{\circ}$ angle. To do that, we must find the perpendicular slope, which is the opposite reciprocal of our slope $\left(m\right) = - \frac{5}{2}$.

Perpendicular slope is $= - \left(- \frac{2}{5}\right) = \frac{2}{5}$

color(green)(ul (Step:2

Use the Point-Slope Formula to find the equation.

Point-slope formula: color(blue)(y=m(x-h)+k, where
$m$ is the perpendicular slope and $\left(h , k\right)$ represent the vertex $C$ at $\left(5 , 2\right)$

Hence, $y = \left(\frac{2}{5}\right) \left(x - 5\right) + 2$

$y = \frac{2}{5} x - \frac{10}{5} + 2$

$y = \frac{2}{5} x$ " "color(red)(Equation.1

color(green)(ul (Step:3

We will repeat the process from color(green)(ul (Step:1 and color(green)(ul (Step:2

Consider side $A C$. Vertices are $A \left(3 , 1\right) \mathmr{and} C \left(5 , 2\right)$

Next, we find the slope.

$m = \frac{2 - 1}{5 - 3}$

$m = \frac{1}{2}$

Find the perpendicular slope.

$= \Rightarrow - \left(\frac{2}{1}\right) = - 2$

color(green)(ul (Step:4

Point-slope formula: color(blue)(y=m(x-h)+k, using the vertex $B$ at $\left(1 , 6\right)$

Hence, $y = \left(- 2\right) \left(x - 1\right) + 6$

$y = - 2 x + 8$ " "color(red)(Equation.2

color(green)(ul (Step:5

Find the solution to the system of linear equations to find the vertices of the Orthocenter of the triangle.

$y = \frac{2}{5} x$ " "color(red)(Equation.1

$y = - 2 x + 8$ " "color(red)(Equation.2#

The solution is becoming too long. Method of Substitution will provide solution for the system of linear equations.

Orthocenter $= \left(\frac{10}{3} , \frac{4}{3}\right)$

The construction of the triangle with the Orthocenter is: