# What is the equation of the line that passes through the origin and is perpendicular to the line that passes through the following points: (3,7),(5,8)?

Jun 29, 2018

$y = - 2 x$

#### Explanation:

First of all, we need to find the gradient of the line passing through $\left(3 , 7\right)$ and $\left(5 , 8\right)$

$\text{gradient} = \frac{8 - 7}{5 - 3}$

$\text{gradient} = \frac{1}{2}$

Now since the new line is PERPENDICULAR to the line passing through the 2 points, we can use this equation

${m}_{1} {m}_{2} = - 1$ where the gradients of two different lines when multiplied should equal to $- 1$ if the lines are perpendicular to one another ie at right angles .

hence, your new line would have a gradient of $\frac{1}{2} {m}_{2} = - 1$
${m}_{2} = - 2$

Now, we can use the point gradient formula to find your equation of the line
$y - 0 = - 2 \left(x - 0\right)$
$y = - 2 x$

Jun 29, 2018

Equation of the passing through the origin and having slope = -2 is

color(blue)(y = -2x " or " 2x + y = 0

#### Explanation:

$A \left(3 , 7\right) , B \left(5 , 8\right)$

$\text{Slope of line AB } = m = \frac{{y}_{b} - {y}_{a}}{{x}_{b} - {x}_{a}} = \frac{8 - 7}{5 - 3} = \frac{1}{2}$

Slope of the perpendicular line = -1/m = -2

Equation of the passing through the origin and having slope = -2 is

$\left(y - 0\right) = - 2 \left(x - 0\right)$

color(blue)(y = -2x " or " 2x + y = 0#

graph{-2x [-10, 10, -5, 5]}