How to find and graph parallel and perpendicular lines

 
 
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How to define parallel and perpendicular lines

In this lesson we’ll learn about the qualities that make up parallel and perpendicular lines and how to identify them on a graph or in an equation.

Remember, opposites are numbers with different signs, as a variable they can be expressed as mm and m-m (although this doesn’t mean that mm is the positive number and m-m is the negative number).

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Here are a few more examples of opposite pairs:

4, 44, \ -4

0.15 0.150.15 \ -0.15

34, 34\frac{3}{4}, \ -\frac{3}{4}

Reciprocals are fractions that, when multiplied together give you 11. Usually we say they are “flipped” numbers. Here are some examples of reciprocals.

12, 2\frac{1}{2}, \ 2

35, 53\frac{3}{5}, \ \frac{5}{3}

27, 72-\frac{2}{7}, \ -\frac{7}{2}

The formula for a line in slope-intercept form is y=mx+by=mx+b, where mm is the slope of the line and bb is the yy-intercept.

Parallel lines

For two lines to be parallel, their slopes must be equal but the yy-intercepts must be different (otherwise they are the same line).

Algebraically, in slope-intercept form they would look like

{y=mx+b1y=mx+b2\begin{cases}y=mx+ b_{1} \\y=mx+b_{2}\end{cases}

where b1b2b_{1} \neq b_{2}.

Here’s an example of what parallel lines could be written as in slope-intercept form.

{y=2x5y=2x+3\begin{cases}y=2x-5 \\y=2x+3\end{cases}

{y=12x5y=12x+3\begin{cases}y=-\frac{1}{2}x-5 \\y=-\frac{1}{2}x+3\end{cases}

These are both examples of parallel line pairs.             

Parallel lines go on forever in the same direction and never cross one another. Here are the two examples of parallel line pairs when they are graphed.

 
pairs of parallel lines
 

Perpendicular lines

Perpendicular lines have opposite reciprocal slopes, so their yy-intercepts can be the same.

Algebraically, in slope-intercept form they would look like

{y=mx+b1y=1mx+b2\begin{cases}y=mx+ b_{1} \\y=-\frac{1}{m}x+b_{2}\end{cases}

where b1b_{1} and b2b_{2} could be the same.

Here are two examples of perpendicular line pairs:

{y=2x3y=12x3\begin{cases}y=2x-3 \\y=-\frac{1}{2}x-3\end{cases}

{y=5x+2y=15x4\begin{cases}y=-5x+2 \\y=\frac{1}{5}x-4\end{cases}

When perpendicular lines are graphed they cross each other and form four 9090 degree angles. Be aware that checking this on your calculator might not always help because the graphing window settings can disguise right angles.

Here are the examples when they are graphed:

 
pairs of perpendicular lines
 

Let’s go ahead and look at a few of the types of questions you could be asked about when it comes to parallel and perpendicular lines.

 
 

Graphing pairs of parallel and perpendicular lines


 
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Finding pairs of parallel and perpendicular lines

Example

Write the equation of the line parallel to 5x+2y=105x+2y=10 with a yy-intercept of 44.

For two lines to be parallel, their slopes must be equal.

Remember that the equation of a line in slope-intercept form is given by

y=mx+by=mx+b

where mm is the slope of the line. We can rewrite the given equation as

5x+2y=105x+2y=10

5x5x+2y=5x+105x-5x+2y=-5x+10

2y=5x+102y=-5x+10

122y=125x+1210\frac{1}{2}\cdot 2y=\frac{1}{2}\cdot -5x+\frac{1}{2}\cdot 10

y=52x+5y=-\frac{5}{2}x+5

The slope of the line is 5/2-5/2.

We want to write an equation of a line with a slope of 5/2-5/2 and a yy-intercept of 44. So m=5/2m=-5/2 and b=4b=4. Therefore, the equation is

y=52x+4y=-\frac{5}{2}x+4


Let’s look at another example.


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For two lines to be parallel, their slopes must be equal but the Y-intercepts must be different (otherwise they are the same line).

Example

Graph the line parallel to the one in the graph with a yy-intercept of 2-2.

single line

Parallel lines go on forever in the same direction and have the same slope. We want to draw a line that never crosses this one but that goes through the yy-intercept of (0,2)(0,-2).

Start by graphing the yy-intercept of 2-2. Find the rise over run (the slope) of the original line from its yy-intercept, then follow the rise over run pattern on the new yy-intercept and draw a point. Connect the points and draw the parallel line.

pair of parallel lines

The new line has an equation of

y=13x2y=-\frac{1}{3}x-2


Let’s look at an example of a perpendicular line instead.


Example

Write the equation of the line passing through (2,5)(-2,5) and perpendicular to

y=47x2y=-\frac{4}{7}x-2

Remember, perpendicular lines have opposite reciprocal slopes. In other words, the slope of the new line needs to be the negative reciprocal of 4/7-4/7, which means the slope of the new line is

74\frac{7}{4}

The equation of a line in slope-intercept form is y=mx+by=mx+b. For our new line we know the slope and a point. The slope is m=7/4m=7/4 and the point is (2,5)(-2,5).

We can plug the slope and point into the formula and solve for the yy-intercept bb of the new line.

5=74(2)+b5=\frac{7}{4}(-2)+b

5=72+b5=-\frac{7}{2}+b

5+72=72+72+b5+\frac{7}{2}=-\frac{7}{2}+\frac{7}{2}+b

172=b\frac{17}{2}=b

Now plug the slope and yy-intercept into the equation. The perpendicular line is

y=74x+172y=\frac{7}{4}x+\frac{17}{2}

 
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