Writing Equations of Perpendicular Lines

a

Parallel lines have the same slopes. Thus, we need to determine the slope of the given line. Let's do that by identifying the coordinates of two points in the line.

With the two points we can find the slope using the slope formula.

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

SubstitutePointsSubstitute

(6, 4)

&

(2, 2)

m = \frac{4 - 2}{6 - 2}

SubTermsSubtract terms

m = \frac{2}{4}

ReduceFrac

\frac{a}{b} = \frac{a / 2}{b / 2}

m = \frac{1}{2}

The parallel line must have the slope

m = \frac{1}{2} .

This means we only have to find the

y

-intercept of the given line to graph it. The

y

-intercept of a parallel line can not be the same as the

y

-intercept of the given line.

We can choose any $y$ -intercept not equal to $(0, 1) .$ Let's go with $(0, 4) .$ Plot that point and another one using the slope. Then complete the graph with a line through the points.

b

The product of the slopes of perpendicular lines is

- 1 .

Put in

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

into the formula

m_{1} \cdot m_{2} = - 1

and solve for the slope of the perpendicular line. This will give us the slope of our new line, the perpendicular one.

m_{1} \cdot m_{2} = - 1

Substitute

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

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

MultEqn

LHS \cdot 2 = RHS \cdot 2

m_{1} = - 2

The line's slope is then

m_{2} = - 2

. The second thing we know about perpendicular lines is that they will intersect in one point. It doesn't matter which point you choose as the point of intersection as long as it is on the given line.

Writing Equations of Perpendicular Lines 1.9 - Solution