ACGC - Parallel and Perpendicular Lines (Lesson)

Parallel and Perpendicular Lines

Parallel Lines

When graphing two different straight lines, they either never cross or they cross one time. Lines that never cross are called parallel lines. Watch the video below to learn more about how we know whether or not two lines are parallel.

Point-Slope Form = $y-y_1=m(x-x_1)$

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This next video runs through a few examples of finding parallel lines when the first line is NOT in slope-intercept form. There are a few extra steps when this is the case. 

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Application: Parallel Lines

Example: A park is being built and the design blueprint is shown below. There is a creek that runs through the park in a straight line. The creek runs through points (0, 2) and (5, 5). The plan is the build a path that follows the creek in a parallel line. The path should run through the point (4, 6).

Find the slope of the line that represents the creek.

$\frac{y_2-y_1}{x_2-x_1}=\frac{5-2}{5-0}=\frac{3}{5}$

The slope is $\frac{3}{5}$.

Now, using this slope find the equation of the parallel line that runs through the point (4,  6).

$y-y_1=m(x-x_1)$

$y-6=\frac{3}{5}(x-4)$             Replace $(x_1,y_1)$ with (4, 6)

$y-6=\frac{3}{5}x-\frac{12}{5}$              Distribute the $\frac{3}{5}$ 

$y=\frac{3}{5}x-\frac{12}{5}+6$              Add 6 to both sides

$y=\frac{3}{5}x-\frac{12}{5}+\frac{30}{5}$           Change the denominator from $\frac{6}{1}$ to $\frac{30}{5}$ so we can add the fractions 

$y=\frac{3}{5}x+\frac{18}{5}$                      Add the fractions

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Perpendicular Lines

Now that we know about parallel lines, we need to talk about perpendicular lines. These are lines that cross one time, at a right angle. Watch the videos below to learn more about perpendicular lines.

Negative Reciprocal -  Swap the numerator and denominator and add a negative sign. Example: $\frac{2}{3}=-\frac{3}{2}$

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This next video runs through a few examples of finding perpendicular lines when the first line is NOT in slope-intercept form. There are a few extra steps when this is the case. 

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Application: Perpendicular Lines

Example: Now that we have our path that runs parallel to the creek, we'd like to add a perpendicular path that crosses the creek (with a bridge of course). Using the information that you found previously, find the equation of the line that is perpendicular to the creek and crosses at the point (4, 6).

Determine the slope of a line that is perpendicular to the line that represents the creek.

We found the slope of the creek in the last example to be $\frac{3}{5}$.  The negative reciprocal of this value is $-\frac{5}{3}$.

Now, using the negative reciprocal find the equation of the perpendicular line that runs through the point (4,  6).

$y-y_1=m(x-x_1)$

$y-6=-\frac{5}{3}(x-4)$            Replace $(x_1,y_1)$ with (4, 6)

$y-6=-\frac{5}{3}x+\frac{20}{3}$             Distribute the $-\frac{5}{3}$ 

$y=-\frac{5}{3}x+\frac{20}{3}+6$             Add 6 to both sides

$y=-\frac{5}{3}x+\frac{20}{3}+\frac{18}{3}$          Change the denominator from $\frac{6}{1}$ to $\frac{18}{3}$ so we can add the fractions 

$y=-\frac{5}{3}x+\frac{38}{3}$                      Add the fractions

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It's time to apply what you've learned! Try the practice problems below to see how much you know!

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