Find the Slope of the Perpendicular Line to the Line Through the Two Points (2,3) , (5,7)

$(2, 3)$ , $(5, 7)$

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.

$m = \frac{7 - (3)}{5 - (2)}$

Step 4

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{7 - 3}{5 - (2)}$

Subtract $3$ from $7$ .

$m = \frac{4}{5 - (2)}$

Simplify the denominator.

Multiply $- 1$ by $2$ .

$m = \frac{4}{5 - 2}$

Subtract $2$ from $5$ .

$m = \frac{4}{3}$

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 $- \frac{1}{\frac{4}{3}}$ .

Multiply the numerator by the reciprocal of the denominator.

$m_{perpendicular} = - (1 (\frac{3}{4}))$

Multiply $\frac{3}{4}$ by $1$ .

$m_{perpendicular} = - \frac{3}{4}$

Step 7

Find the Slope of the Perpendicular Line to the Line Through the Two Points (2,3) , (5,7)

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