10.3: Parallel and Perpendicular
Learning Objectives
- Identify and compute slope in the coordinate plane.
- Use the relationship between slopes of parallel lines.
- Use the relationship between slopes of perpendicular lines.
- Identify equations of parallel lines.
- Identify equations of perpendicular lines.
Slope in the Coordinate Plane
If you look at a graph of a line, you can think of the slope as the steepness of the line.
- Slope is the measure of the ______________________________ of a line.
Mathematically, you can calculate the slope using two different points on a line. Given two points \begin{align*}(x_1, y_1)\end{align*} and \begin{align*}(x_2, y_2)\end{align*} the slope is:
\begin{align*}slope = \frac{y_2 - y_1}{x_2 - x_1}\end{align*}
You may have also learned that slope equals “rise over run.”
This means that:
- The numerator (top) of the fraction is the “rise,” or how many units the slope goes up (positive) or down (negative).
\begin{align*}\rightarrow\end{align*} Up or down is how the slope moves along the \begin{align*}y-\end{align*}axis.
- The denominator (bottom) of the fraction is the “run,” or how many units the slope goes to the right (positive) or left (negative).
\begin{align*}\rightarrow\end{align*} Right or left is how the slope moves along the \begin{align*}x-\end{align*}axis.
You can remember “rise” as up or down because an elevator “rises” up or down.
“Rise” (up/down) is in the \begin{align*}y\end{align*} direction.
You can remember “run” as moving right or left because a person “runs” with her right and left feet.
“Run” (right/left) is in the \begin{align*}x\end{align*} direction.
- The numerator of the slope represents the change in the _________ direction.
- The slope’s _______________________ represents the change in the \begin{align*}x\end{align*} direction.
In other words, first calculate the distance that the line travels up (or down), and then divide that value by the distance the line travels left to right.
A line that goes up from left to right has positive slope, and a line that goes down from left to right has negative slope:
images from http://www.tutorvista.com/math/positive-and-negative-slope
- A line that goes up from left to right has a _________________________ slope.
- A line that goes down from left to right has a ________________________ slope.
Example 1
What is the slope of a line that goes through the points (2, 2) and (4, 6)?
You can use the slope formula on the previous page to find the slope of this line. When substituting values, \begin{align*}(x_1, y_1)\end{align*} is (2, 2) and \begin{align*}(x_2, y_2)\end{align*} is (4, 6):
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}, y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;},\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}, y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope &= \frac{y_2 - y_1}{x_2 - x_1}\\ slope &= \frac{6 - 2}{4 - 2} = \frac{4}{2} = 2\end{align*}
The slope of this line is 2.
\begin{align*}\rightarrow\end{align*} What does that mean graphically?
Look at the graph on the next page to see what the line looks like.
Notice: If the slope is positive, the line should go up from left to right. Does it?
You can see that the line “rises” 4 units as it “runs” 2 units to the right. So, the “rise” (numerator) is 4 units and the “run” (denominator) is 2 units. Since \begin{align*}4 \div 2 = 2\end{align*}, the slope of this line is 2.
As you read on the previous page, the slope of this line is 2, a positive number.
- Any line with a positive slope will go ____________________ from left to right.
- Any line with a negative slope will go ____________________ from left to right.
Example 2
What is the slope of the line that goes through the points (1, 9) and (3, 3)?
Again, use the formula to find the slope of this line:
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope &= \frac{y_2 - y_1}{x_2 - x_1}\\ slope &= \frac{3 - 9}{3 - 1} = \frac{-6}{2} = -3\end{align*}
The slope of this line is –3.
Because the slope of the line in example 2 is negative, it will go down to the right. The points and the line that connects them are shown below:
Some types of lines have special slopes. Check out following examples to see what happens with horizontal and vertical lines.
Example 3
What is the slope of a line that goes through the points (4, 4) and (8, 4)?
Use the formula to find the slope of this line:
\begin{align*}x_1 = \underline{\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;},\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;}\end{align*}
\begin{align*}slope & = \frac{y_2 - y_1}{x_2 - x_1}\\ slope & = \frac{4 - 4}{8 - 4} = \frac{0}{8}= 0\end{align*}
The slope of this line is 0.
Every line with a slope of 0 is horizontal.
- A ____________________________________ line has a slope equal to zero.
Example 4
What is the slope of a line that goes through the points (3, 2) and (3, 6)?
Use the formula to find the slope of this line:
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope & = \frac{y_2 - y_1}{x_2 - x_1}\\ slope & = \frac{6 - 2}{3 - 3} = \frac{4}{0}\end{align*}
Zero is not allowed to be in the denominator of a fraction! Therefore, the slope of this line is undefined.
Every line with an undefined slope is vertical.
- All vertical lines have slopes that are ____________________.
The line in example 4 is vertical and its slope is undefined:
In review, if you look at a graph of a line from left to right, then:
- Lines with positive slopes point up to the right.
- Lines with negative slopes point down to the right.
- Horizontal lines have a slope of zero.
- Vertical lines have undefined slope.
Reading Check:
1. On the coordinate plane below, draw a line with a positive slope.
2. On the coordinate plane below, draw a line with a negative slope.
3. On the coordinate plane below, draw a line with a slope of zero.
4. On the coordinate plane below, draw a line with an undefined slope.
Slopes of Parallel Lines
Now that you know how to find the slope of lines using \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinates, you can think about how lines are related to their slopes.
If two lines in the coordinate plane are parallel, then they will have the same slope. Conversely, if two lines in the coordinate plane have the same slope, then those lines are parallel.
- Parallel lines have the _____________________________ slope.
Example 5
Which of the following answers below could represent the slope of a line parallel to the one shown on the graph?
A. – 4
B. – 1
C. \begin{align*}\frac{1}{4}\end{align*}
D. 1
Since you are looking for the slope of a parallel line, it will have the same slope as the line on the graph. First find the slope of the given line, and then choose the answer with that same slope. To do this, pick any two points on the line and use the slope formula.
For example, for the points (–1, 5) and (3, 1) :
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope & = \frac{y_2 - y_1}{x_2 - x_1}\\ slope & = \frac{1 - 5}{3 - (-1)} = \frac{-4}{3 + 1} = \frac{-4}{4} = -1\end{align*}
The slope of the line on the graph is –1. The answer is \begin{align*}B\end{align*}.
Slopes of Perpendicular Lines
Parallel lines have the same slope. There is also a mathematical relationship for the slopes of perpendicular lines.
The slopes of perpendicular lines will be the opposite reciprocal of each other.
Opposite here means the opposite sign.
If a slope is positive, then its opposite is negative.
If a slope is negative, then its opposite is positive.
A reciprocal is a fraction with its numerator and denominator flipped.
The reciprocal of \begin{align*}\frac{2}{3}\end{align*} is \begin{align*}\frac{3}{2}\end{align*}. The reciprocal of \begin{align*}\frac{1}{2}\end{align*} is 2. The reciprocal of 4 is \begin{align*}\frac{1}{4}\end{align*}.
- The opposite of 5 is _______________.
- The reciprocal of 5 is _______________.
The opposite reciprocal can be found in two steps:
1. First, find the reciprocal of the given slope. If the slope is a fraction, you can simply switch the numbers in the numerator and the denominator to find the reciprocal. If the slope is not a fraction, you can make it into a fraction by putting a 1 in the denominator. Then find the reciprocal by flipping the numerator and denominator.
2. The second step is to find the opposite of the given number. If the value is positive, make it negative. If the value is negative, make it positive.
The opposite reciprocal of \begin{align*}\frac{5}{4}\end{align*} is \begin{align*}- \frac{4}{5}\end{align*} and the opposite reciprocal of - 3 is \begin{align*}\frac{1}{3}\end{align*}.
- The opposite reciprocal of 5 is _______________.
Another way to check if lines are perpendicular is to multiply their slopes: if the slopes of two lines multiply to be –1, then the two lines are perpendicular.
- The slopes of ____________________________ lines multiply to be –1.
Example 6
Which of the following numbers could represent the slope of a line perpendicular to the one shown below?
A. \begin{align*}\frac{-7}{5}\end{align*}
B. \begin{align*}\frac{7}{5}\end{align*}
C. \begin{align*}\frac{-5}{7}\end{align*}
D. \begin{align*}\frac{5}{7}\end{align*}
Since you are looking for the slope of a perpendicular line, it will be the opposite reciprocal of the slope of the line on the graph. First find the slope of the given line, then find its opposite reciprocal. You can use the slope formula to find the original line’s slope. Pick two points on the line.
For example, for the points (–3, –2) and (4, 3) :
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_ 2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope &= \frac{y_2 - y_1}{x_2 - x_1}\\ slope &= \frac{3 - (-2)}{4 - (-3)} = \frac{3 + 2}{4 + 3} = \frac{5}{7}\end{align*}
The slope of the line on the graph is \begin{align*}\frac{5}{7}\end{align*}. Now find the opposite reciprocal of that value. First switch the numerator and denominator in the fraction, then find the opposite sign. The opposite reciprocal of \begin{align*}\frac{5}{7}\end{align*} is \begin{align*}\frac{-7}{5}\end{align*}. The answer is A.
Slope-Intercept Equations
The most common type of linear equation to study is called slope-intercept form, which uses both the slope of the line and its \begin{align*}y-\end{align*}intercept. A \begin{align*}y-\end{align*}intercept is the point where the line crosses the vertical \begin{align*}y-\end{align*}axis. This is the value of \begin{align*}y\end{align*} when \begin{align*}x\end{align*} is equal to 0.
- Slope-intercept form is an equation that uses the _________________________ and the ______________________________ of a line.
- The \begin{align*}y-\end{align*}intercept is the point where the line intersects the ___________________.
- At the \begin{align*}y-\end{align*}intercept, \begin{align*}x\end{align*} equals ______________.
The formula for an equation in slope-intercept form is:
\begin{align*}y = mx + b\end{align*}
In this equation, \begin{align*}y\end{align*} and \begin{align*}x\end{align*} remain as variables, \begin{align*}m\end{align*} is the slope of the line, and \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}intercept of the line. For example, if you know that a line has a slope of 4 and it crosses the \begin{align*}y-\end{align*}axis at (0, 8), then its equation in slope-intercept form is: \begin{align*}y = 4x + 8\end{align*}.
- In slope-intercept form, \begin{align*}m\end{align*} represents the _____________.
- In slope-intercept form, \begin{align*}b\end{align*} represents the _____________.
This form is especially useful for finding the equation of a line given its graph. You already know how to calculate the slope by finding two points and using the slope formula. You can find the \begin{align*}y-\end{align*}intercept by seeing where the line crosses the \begin{align*}y-\end{align*}axis on the graph. The value of \begin{align*}b\end{align*} is the \begin{align*}y-\end{align*}coordinate of this point.
Example 7
Write an equation in slope-intercept form that represents the following line:
First find the slope of the line. You already know how to do this using the slope formula. There are no points given on the line, so you have to pick your own points. See where the line goes right through an intersection (corner point) on the graph paper. You can use the two points (0, 3) and (2, 2) :
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}, y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope &= \frac{y_2 - y_1}{x_2 - x_1}\\ slope &= \frac{2 - 3}{2 - 0} = \frac{-1}{2} = - \frac{1}{2}\end{align*}
The slope of the line is \begin{align*}\frac{-1}{2}\end{align*}. This will replace \begin{align*}m\end{align*} in the slope-intercept equation.
Now you need to find the \begin{align*}y-\end{align*}intercept. On the graph, find where the line intersects the \begin{align*}y-\end{align*}axis. It crosses the \begin{align*}y-\end{align*}axis at (0, 3) so the \begin{align*}y-\end{align*}intercept is 3. This will replace \begin{align*}b\end{align*} in the slope-intercept equation, so now you have all the information you need.
The equation for the line shown in the graph is: \begin{align*}y = - \frac{1}{2} x + 3\end{align*}.
- In the slope-intercept equation, the slope is represented by the letter _________.
- In the slope-intercept equation, the \begin{align*}y-\end{align*}intercept is the letter _________.
Equations of Parallel Lines
You studied parallel lines and their graphical relationships, so now you will learn how to easily identify equations of parallel lines. When looking for parallel lines, look for equations that have the same slope.
As long as the \begin{align*}y-\end{align*}intercepts are not the same and the slopes are equal, the lines are parallel. If the \begin{align*}y-\end{align*}intercept and the slope are both the same, then the two equations are for the same exact line, and a line cannot be parallel to itself.
- Parallel lines have the _______________________ slope.
Reading Check:
1. True or false:
The reciprocal of a fraction is when you flip the numerator and the denominator.
2. Make up an example that supports the statement in #1 above.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. What is the slope-intercept form of an equation?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
4. What do the letters \begin{align*}m\end{align*} and \begin{align*}b\end{align*} stand for in the slope-intercept equation?
\begin{align*}m :\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}b :\end{align*}
\begin{align*}{\;}\end{align*}
5. How are the slopes of parallel lines related?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Example 8
Juan drew the line below:
Which of the following equations could represent a line parallel to the one Juan drew?
A. \begin{align*}y = -\frac{1}{2} x - 6\end{align*}
B. \begin{align*}y = \frac{1}{2} x + 9 \end{align*}
C. \begin{align*}y = -2x - 18\end{align*}
D. \begin{align*}y = 2x + 1\end{align*}
If you find the slope of the line in Juan’s graph, you can find the slope of a parallel line because it will be the same. Pick two points on the graph and find the slope using the slope formula. Use the points (0, 5) and (1, 3) :
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope &= \frac{y_2 - y_1}{x_2 - x_1}\\ slope &= \frac{3 - 5}{1 - 0} = \frac{- 2}{1} = - 2\end{align*}
The slope of Juan’s line is –2. Look at your four answer choices: which equation has a slope of –2? All other parts of the equation do not matter. The only equation that has a slope of –2 is choice C, so that is the correct answer.
Equations of Perpendicular Lines
You also studied perpendicular lines and their graphical relationships: remember that the slopes of perpendicular lines are opposite reciprocals. To easily identify equations of perpendicular lines, look for equations that have slopes that are opposite reciprocals of each other.
Here, it does not matter what the \begin{align*}y-\end{align*}intercept is; as long as the slopes are opposite reciprocals, the lines are perpendicular.
Example 9
Kara drew the line in this graph:
Which of the following equations could represent a line perpendicular to the one Kara drew above?
A. \begin{align*}y = \frac{3}{2} x + 10\end{align*}
B. \begin{align*}y = -\frac{3}{2} x + 6\end{align*}
C. \begin{align*}y = \frac{2}{3} x - 4\end{align*}
D. \begin{align*}y = -\frac{2}{3} x - 1\end{align*}
First find the slope of the line in Kara’s graph. Then find the opposite reciprocal of this slope. To begin, pick two points on the graph and calculate the slope using the slope formula. Use the points (0, 2) and (3, 4) :
\begin{align*}x_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_1 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} ,\end{align*} and \begin{align*}x_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;} , y_2 = \underline{\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;\; \; \; \; \;}\end{align*}
\begin{align*}slope &= \frac{y_2 - y_1}{x_2 - x_1}\\ slope &= \frac{4 - 2}{3 - 0} = \frac{2}{3}\end{align*}
The slope of Kara’s line on the graph is \begin{align*}\frac{2}{3}\end{align*}.
Find the opposite reciprocal: the reciprocal of \begin{align*}\frac{2}{3}\end{align*} is \begin{align*}\frac{3}{2}\end{align*}, and the opposite of \begin{align*}\frac{3}{2}\end{align*} is \begin{align*}- \frac{3}{2}\end{align*}.
So, \begin{align*}-\frac{3}{2}\end{align*} is the opposite reciprocal of (or perpendicular slope to) \begin{align*}\frac{2}{3}\end{align*}.
Now look in your answer choices for the equation that has a slope of \begin{align*}-\frac{3}{2}\end{align*}.
The only equation that has a slope of \begin{align*}-\frac{3}{2}\end{align*} is choice B, so that is the correct answer.
Reading Check:
1. How are the slopes of perpendicular lines related to each other?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. In the context of perpendicular slope values, what does opposite mean?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. True or false: On a graph, perpendicular lines intersect at an angle of \begin{align*}45^\circ\end{align*}.
4. Correct the statement in #3 above to make it true.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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