Slopes of parallel and perpendicular lines
Key notes:
What is slope?
The slope (mmm) of a line measures how steep the line is.
Formula:
m=y2−y1 / x2−x1
where (x1,y1) and (x2,y2)) are two points on the line.
Slopes of Parallel Lines
Definition: Two lines are parallel if they never meet and are always the same distance apart.
Rule:
Parallel lines have the same slope (if they are not vertical).
m1=m2
Example:
- Line 1: y=2x+3→ slope m1=2
- Line 2: y=2x−5 → slope m2=2
Since m1=m2, the lines are parallel.
Slopes of Perpendicular Lines
Definition: Two lines are perpendicular if they meet at a right angle (90°).
Rule:
The slopes of perpendicular lines are negative reciprocals of each other (if neither is vertical).
m1×m2=−1
Example:
- Line 1: y=1/2x+4→ slope m1=1/2
- Perpendicular line: slope m2=−2
(because 1/2×−2=−1)
Special Cases
- Horizontal line: slope m=0
- Vertical line: slope is undefined
- A horizontal line is perpendicular to a vertical line.
Summary Table
| Relationship | Slope Condition | Example Slopes |
|---|---|---|
| Parallel lines | m1=m2 | 3 and 3 |
| Perpendicular lines | m1×m2=−1 | 2/3 and −3/2 |
Practice Questions
- Are the lines y=4x+1 and y=4x−7 parallel or perpendicular?
- Find the slope of a line parallel to y=−3/5x+2
- Find the slope of a line perpendicular to y=7x−4
- Determine whether the lines with slopes 5 and −1/5 are parallel or perpendicular.
Learn with an example
Line e has a slope of -9/7. line f is perpendicular to e.
what is the slope of line f?
Line f is perpendicular to line e, so its slope is the opposite reciprocal. Find the opposite reciprocal.
-9/7 —–>Take the slope of line e
-7/9 ——> Find the reciprocal
7/9 ——-> Find the opposite
| The slope of line f is 7/9 |
Line u has a slope of 5/9. Line v is parallel to line u.
What is the slope of line v?
Line v is parallel to u, so its slope is the same.
The slope of line v is 5/9.
Line u has a lope of 6/5. Line v is parallel to line u.
What is the slope of line v?
Line v is parallel to u, so its slope is the same.
The slope of line v is 6/5.
Let’s practice:
