Parallel and Perpendicular Lines
Key Concepts
- Write an equation of a line parallel to given line
- Understand the slope of perpendicular lines
- Write an equation of a line perpendicular to a given line
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect.
Perpendicular Lines
Perpendicular lines are lines that intersect at a right (90 degrees) angle.
Write an equation of a line Parallel to given line
Example 1:
What is the equation of the line in slope intercept form that passes through the point (-3, 5) and is parallel to the graph of y=4x -7?
Solution:
Step 1:
Identify the slope of the given line.
y = 4x – 7
The slope is 4. The slope of a parallel line will be the same.
Step 2:
Start with point form. Use the given point and the slope of the parallel line.
y- 5 = 4 (x+3)
y-5 = 4x +12
y= 4x +17
The equation of the line is y= 4x +17.
Understand the slope of perpendicular lines
Why does it make sense that the slopes of perpendicular lines have opposite signs?
Solution:
Perpendicular lines are a bit more complicated.
If you visualize a line with a positive slope (so it’s an increasing line), then the perpendicular line must have a negative slope (because it will have to be a decreasing line).
So, perpendicular lines have slopes which have opposite signs.
Example:
Find the slope of a line perpendicular to the line y = −4x + 9.
They’ve given me the original line’s equation, and it’s in “y =” form, so it’s easy to find the slope.
I can just read the value of the equation: m = −4.
This slope can be turned into a fraction by putting it over 1, so this slope can be restated as:
To get the negative reciprocal, I need to flip this fraction and change the sign.
Then the slope of any line perpendicular to the given line is:
Write an equation of a line perpendicular to a given line
Example: What is the equation to the line that passes through (6, -5) and is perpendicular to the graph of y=2x+3?
Solution:
Step 1:
Use the slope of the given line to determine the slope of the line that is perpendicular.
y=2x+3.
m = 2
The slope of a line perpendicular to the given line is the opposite reciprocal of 2 / 1
Use −1 / 2 as the slope of new line.
Step 2: Start with the point-slope form. Use the given point and slope of the perpendicular line.
y+5 = −1 / 2 (x – 6)
The graph of y + 5 = −1 / 2 (x – 6) passes through the point (6, -5) and is perpendicular to the graph of y=2x+3.
Exercise
- What is the equation of the line parallel to y=3x+5 and through the point (1, 7)?
- What is the equation of the line parallel to y=4x+3 and through the point (5, 9)?
- What is the equation of the line that is perpendicular to y=2x+10 and goes through the point (5, 1)?
- What is the equation of the line parallel to y= 3/4x+1 and through the point (-4, 9)?
- Determine the equation of a line perpendicular to y=3x−2 at the point (2, 4).
- What is the equation of the line that passes through (4, 5) and is perpendicular to the graph of y=2x-3?
- Define parallel lines.
- Define perpendicular lines.
- Are the graph of the equations 4y=2x-5 and y=-2x +7 parallel, perpendicular, or neither?
- Write the equation of a line that is perpendicular to y=−1/2 x+4 and goes through the point (0, 6)?
Concept Map
What have we learned
- Understand parallel and perpendicular lines
- Write an equation of a line Parallel to given line
- Understand the slope of perpendicular lines
- Understand how to write an equation of a line perpendicular to a given line.
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