Chapter 3 Section 7
Slopes of Parallel and Perpendicular Lines
How to determine if two lines are parallel
If two lines are parallel, then their slopes will always be EQUAL.
How to determine if two lines are perpendicular
If two lines are parallel, then the product of their slopes will be -1
You will usually be given either a graph with two lines or 2 pairs of coordinates. When you are given a graph with two lines, you must find two coordinates on each line. After you have found the coordinates, take one line and find that slope of it using the coordinates you have just found. Do the same for the other line. If they are the same, then they are parallel.
Example: I was given two lines and I have found 2 coordinate pairs, 1 on each line. On the first line I chose (1,5) and (-2, -4). The slope is 3. Now I move onto the next line and the coordinates are (3,3) and (1,-4). The slope is 3 ½, which is not equal to 3 so therefore these lines are not parallel.
If you just given pairs of coordinates, you can just find the slopes of each pair
Example: The coordinate pair of a line is (0, 3) and (4, 1). Subtract 3 and 1, the answer is 2, then subtract 0 and 4 which will be -4. So the slope will be 1/-4. The coordinate pair of the next line is (-6, 1) and (0, -2). Again, subtract 1 from -2 which will be 3, then -6 and 0 which is still -6. The slope, 3/ -6 simplifies to 1/-2 and that is not equal to 1/-4, so they are not parallel.
For perpendicular lines you will also be given either a graph for 3 coordinate pairs. In order to find out if they are perpendicular, you must find two coordinates on each line. After you have found the coordinates, take one line and find that slope of it using the coordinates you have just found. Do the same for the other line. Then you must multiply the two slopes together, if the product is -1, then the lines are perpendicular.
Example: Find 2 coordinate pairs, 1 on each line. I have chosen (-2, 3) and (6, -3). The slope of this line is -3/4. Then I have chosen (0, 2) and (-3, -2). The slope turns out to be 4/3. Then I multiply -3/4 and 4/3, the product is -1, therefore the two lines are perpendicular.
Writing equations for parallel lines
When you are writing an equation for a parallel line, an equation will usually be given and then it will be asked to draw another line that is parallel through a given point.
Step 1: Figure out what the slope of the given equation is
Step 2: Use point- slope form to write another equation with the given point
-Point slope form is y-y1=m(x-x1)
Example: Write an equation for the line parallel to y=-4x+3 that contains (1, -2)
Step 1: y=-4x+3
↑
Slope
Step 2: y-y1=m(x-x1)
y+2=-4(x-1)
y+2=-4x+4
y=-4x+6
Writing equations for perpendicular lines
When you are writing an equation for a parallel line, an equation will usually be given and then it will be asked to draw another line that is perpendicular through a given point.
Step 1: Identify the slope of the line
Step 2: Set up an equation using the slope you just found and having it equal -1 since the product of perpendicular lines are -1
xm=-1
Step 3: use point-slope form to write another equation with the given point
Example: Write an equation for the line through (-3, 7) and perpendicular to y= -3x-5
Step 1: y= -3x-5
↑
Slope
Step 2: -3m=-1
m=1/3
Step 3: y-y1=m(x-x1)
y-7=1/3(x+3)
y-7=1/3x+1
y=1/3x+8
Practice
Find the slope of each line and determine if they are parallel or not
Find the slope of each line and determine if they are perpendicular or not
4. 5.
6. Write an equation for the line parallel to y=-x+4 that contains (-2, 5)
7. Write an equation for the line parallel to y=-2x+1 that contains (0,3)
8. Write an equation for the line through (6,6) and perpendicular to y=2/3x
9. Write an equation for the line through (4,0) and perpendicular to y=1/2x-5
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Answers
- Slope of line 1 is -1/2. Slope of line 2 is -1/2. They are parallel.
- Slope of line 1 1/3. Slope of line 2 is ½. They are not parallel
- Slope of line 1 is 3/2. Slope of line 2 is 2. They are not parallel
- Slope of line 1 is -1/2. Slope of line 2 is 2. -1/2 multiplies by 2 is -1. They are perpendicular lines
- Slope of line 1 is -1. Slope of line is 1. -1 multiplied by 1 is -1. They are perpendicular lines.
- y=-x+3
- y= -2x+3
- y= -3/2+15
- y=-2x+8