Find the Slope of the Perpendicular Line to the Line Through the Two Points (-2,2) , (0,-2)

$(- 2, 2)$ , $(0, - 2)$

Step 1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- 2 - (2)}{0 - (- 2)}$

Step 4

Simplify.

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2 - (2)$ and $0 - (- 2)$ .

Rewrite $- 2$ as $- 1 (2)$ .

$m = \frac{- 1 \cdot 2 - (2)}{0 - (- 2)}$

Factor $- 1$ out of $- 1 (2) - (2)$ .

$m = \frac{- 1 (2 + 2)}{0 - (- 2)}$

Reorder terms.

$m = \frac{- 1 (2 + 2)}{0 - 2 \cdot - 1}$

Factor $2$ out of $- 1 (2 + 2)$ .

$m = \frac{2 (- 1 (1 + 1))}{0 - 2 \cdot - 1}$

Cancel the common factors.

Factor $2$ out of $0$ .

$m = \frac{2 (- 1 (1 + 1))}{2 (0) - 2 \cdot - 1}$

Factor $2$ out of $- 2 \cdot - 1$ .

$m = \frac{2 (- 1 (1 + 1))}{2 (0) + 2 (1)}$

Factor $2$ out of $2 (0) + 2 (- - 1)$ .

$m = \frac{2 (- 1 (1 + 1))}{2 (0 + 1)}$

Cancel the common factor.

$m = \frac{2 (- 1 (1 + 1))}{2 (0 + 1)}$

Rewrite the expression.

$m = \frac{- 1 (1 + 1)}{0 + 1}$

Add $1$ and $1$ .

$m = \frac{- 1 \cdot 2}{0 + 1}$

Simplify the denominator.

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1 \cdot 2}{0 + 1}$

Add $0$ and $1$ .

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

Simplify the expression.

Multiply $- 1$ by $2$ .

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

Divide $- 2$ by $1$ .

$m = - 2$

Step 5

The slope of a perpendicular line is the negative reciprocal of the slope of the line that passes through the two given points.

$m_{perpendicular} = - \frac{1}{m}$

Step 6

Simplify $- \frac{1}{- 2}$ .

Move the negative in front of the fraction.

$m_{perpendicular} = \frac{1}{2}$

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

$m_{perpendicular} = 1 (\frac{1}{2})$

Multiply $\frac{1}{2}$ by $1$ .

$m_{perpendicular} = \frac{1}{2}$

Step 7

Find the Slope of the Perpendicular Line to the Line Through the Two Points (-2,2) , (0,-2)

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