# Given that (1, -2), (0, 5) and (-3, 1) are the vertices of a right triangle, how do you determine an equation of the line that passes through the endpoints of the hypotenuse?

Oct 2, 2017

$y = - 7 x + 5.$

#### Explanation:

Let the gven points be, $A \left(1 , - 2\right) , B \left(0 , 5\right) , \mathmr{and} , C \left(- 3 , 1\right) .$

The slope ${m}_{1}$ of line $A B ,$ is, $\frac{- 2 - 5}{1 - 0} = - 7.$

The slope ${m}_{2}$ of line $B C ,$ is, $\frac{1 - 5}{- 3 - 0} = \frac{4}{3.}$

The slope ${m}_{3}$ of line $C A ,$ is, $\frac{1 - \left(- 2\right)}{- 3 - 1} = - \frac{3}{4.}$

$\because {m}_{2} \times {m}_{3} = - 1 , \therefore , B C \bot C A .$

$\therefore \angle C$ is the right angle.

$\therefore A B$ is the hypotenuse.

Using the Slope-Intercept Form, we get,

$y = - 7 x + 5 ,$ as the desired eqn. of the hypo.