# In the coordinate plane, n passes through the points (-1,4) and (5,2) and m passes through the points (2,1) and (4,y). For what value of y is n perpendicular to m?

Jul 4, 2017

See the explanation below.

#### Explanation:

If two lines are perpendicular to each other, their slopes are opposite reciprocals.

To solve this problem, find the slope of line $n$, find the slope that line $m$ should have, and then use the slope to figure out the point $y$.

Slope of line $n$:

m = (Deltax)/(Delta y) = (y_2-y_1)/(x_2-x_1) = (2-4)/(5-(-1)) = (-2)/6 = -1/3

The slope of line $m$ is the opposite reciprocal of $- \frac{1}{3}$, which is $3$.

Using the slope formula, substitute 3 in for the slope and calculate the value of $y$.

$m = \frac{\Delta x}{\Delta y} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{y - 1}{4 - 2} = \frac{y - 1}{2}$

$3 = \frac{y - 1}{2}$

$6 = y - 1$

$y = 7$