Find the Slope of the Perpendicular Line to the Line Through the Two Points (-4,2) , (0,0)

$(- 4, 2)$ , $(0, 0)$

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

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

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).

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

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

$m = \frac{0 - (2)}{0 - (- 4)}$

Simplify.

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Reduce the expression by cancelling the common factors.

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Cancel the common factor of $0 - (2)$ and $0 - (- 4)$ .

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Rewrite $0$ as $- 1 (0)$ .

$m = \frac{- 1 \cdot 0 - (2)}{0 - (- 4)}$

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

$m = \frac{- 1 (0 + 2)}{0 - (- 4)}$

Reorder terms.

$m = \frac{- 1 (0 + 2)}{0 - 4 \cdot - 1}$

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

$m = \frac{2 (- 1 (0 + 1))}{0 - 4 \cdot - 1}$

Cancel the common factors.

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Factor $2$ out of $0$ .

$m = \frac{2 (- 1 (0 + 1))}{2 (0) - 4 \cdot - 1}$

Factor $2$ out of $- 4 \cdot - 1$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Add $0$ and $1$ .

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

Simplify the denominator.

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Multiply $- 2$ by $- 1$ .

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

Add $0$ and $2$ .

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

Simplify the expression.

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Multiply $- 1$ by $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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Cancel the common factor of $1$ and $- 1$ .

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = 1 \cdot 2$

Multiply $2$ by $1$ .

$m_{perpendicular} = - (- 1 \cdot 2)$

Multiply $- 1$ by $2$ .

$m_{perpendicular} = 2$

Multiply $- 1$ by $- 2$ .

$m_{perpendicular} = 2$

Find the Slope of the Perpendicular Line to the Line Through the Two Points (-4,2) , (0,0)