10.01 Equations of parallel and perpendicular lines
Parallel lines
Parallel lines are lines in the same plane that do not intersect. For this to happen the two lines need to have exactly the same slope. If they have different slopes they will cross exactly once.
Recall that we have three ways to present the equation of a straight line:
$y=mx+b$y=mx+b (slope-intercept form)
$Ax+By=C$Ax+By=C (standard form)
$\left(y-y_1\right)=m\left(x-x_1\right)$(y−y1)=m(x−x1) (point-slope form)
If two (or more) lines are parallel then:
- If they are of the form, $y=mx+b$y=mx+b, then the $m$m values are equal.
- If they are of the form, $Ax+By=C$Ax+By=C, then the values $-\frac{A}{B}$−AB are equal (or just rearrange into slope-intercept form and compare $m$m values).
For every straight line $y=mx+b$y=mx+b, there exist infinitely many lines parallel to it.
Here is the line $y=x$y=x
Here are two more lines in the same family of parallel lines.
$y=x+1$y=x+1 and $y=x-1$y=x−1
Same slope ($m$m values)
Different $y$y-intercepts ($b$b values)
- Lines in the same plane that do not intersect are parallel.
- Parallel lines have the same slopes and different $y$y-intercepts.
- All vertical lines are parallel.
Practice questions
Question 1
If the line formed by equation $1$1 is parallel to the line formed by equation $2$2, fill in the missing value below.
Equation $1$1: $y$y$=$=$7x-5$7x−5
Equation $2$2: $y$y$=$= $\editable{}$ $x$x$+$+$2$2
Question 2
Calculate the slopes of lines $AB$AB and $CD$CD, where $A$A, $B$B, $C$C and $D$D have the coordinates:
$A$A $\left(2,5\right)$(2,5)
$B$B $\left(-2,9\right)$(−2,9)
$C$C $\left(-1,3\right)$(−1,3)
$D$D $\left(-7,3\right)$(−7,3)
First, calculate the slope of the line $AB$AB.
Now, find the slope of the line $CD$CD.
Is the line $CD$CD parallel to the line $AB$AB?
Yes
ANo
B
Question 3
Find the equation of the straight line that passes through the point $\left(1,5\right)$(1,5) and is parallel to the straight line with equation $y=-6x-3$y=−6x−3
What is the slope of the new line?
$m$m$=$= $\editable{}$
Now, if the equation has the form $y=mx+b$y=mx+b, solve to find the value of $b$b.
Hence, write the equation of the straight line that passes through the point $\left(1,5\right)$(1,5) and is parallel to the straight line with equation $y=-6x-3$y=−6x−3.
You may express the equation in slope intercept or standard form.
Perpendicular lines
Recall that perpendicular lines are lines that meet at a right ($90^\circ$90°) angle. As you explore the properties of perpendicular lines using the applet below, think about your answer to the guiding questions. You may wish to discuss your answers with a classmate.
Guiding questions
- What do you notice about the slopes of perpendicular lines?
- Change the value of $b$b using the slider. What happens to the graphs?
- Change the value of $m$m using the slider. What happens to the graphs?
- What must change within the equations of perpendicular lines to make them parallel lines?
- Given the equation of two lines, what parameter determines whether they are perpendicular, parallel or neither?
You may continue to investigate, using the applet above creating pairs of perpendicular lines.
Fill in this table as you go.
| Slope of line $1$1 ($m_1$m1) | ||||
| Slope of line $2$2 ($m_2$m2) | ||||
| Product of slopes of lines $1$1 and $2$2 ($m_1\times m_2$m1×m2) |
What do you notice about the product of the slopes of lines $1$1 and $2$2? (The pair of perpendicular lines)
You will have discovered the perpendicular lines have slopes whose product is equal to $-1$−1.
We say that $m_1$m1 is the negative reciprocal of $m_2$m2, if $m_1\times m_2=-1$m1×m2=−1.
Negative reciprocal means they have opposite signs and they are reciprocals of each other.
Here are some examples:
$2$2 and $\frac{-1}{2}$−12
$\frac{3}{4}$34 and $\frac{-4}{3}$−43
$-10$−10 and $\frac{1}{10}$110
- Lines that intersect to form a right angle are perpendicular.
- Perpendicular lines (not parallel to either of the axes) have slopes whose product is $-1$−1.
- Horizontal and vertical lines are perpendicular.
Practice questions
Question 4
If equation $1$1 is perpendicular to equation $2$2, fill in the missing value below.
Equation 1: $y=-10x+4$y=−10x+4
Equation 2: $y$y$=$= $\editable{}$ $x-6$x−6
Question 5
Answer the following.
Line $1$1 has equation $-7x=3y+6$−7x=3y+6. Find its slope.
Line $2$2 has equation $-7y-3x-3=0$−7y−3x−3=0. Find its slope.
Are the two lines perpendicular?
no
Ayes
B
Question 6
Find the equation of a line that is perpendicular to $y=-\frac{3x}{2}+6$y=−3x2+6, and goes through the point $\left(0,2\right)$(0,2).
You may express the equation in slope intercept or standard form.