Parallel lines have equal slopes and different intercepts. Equal slopes with the same intercept is one line.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Perpendicular
Perpendicular slopes are negative reciprocals: flip the fraction and change the sign.
A quick test
Two slopes are perpendicular exactly when their product is −1.
The awkward pair
A horizontal and a vertical line are perpendicular, but the product test cannot be applied — one has no slope.
The slope conditions
Parallel lines have equal slopes. Perpendicular lines have slopes whose product is −1, so each is the negative reciprocal of the other: 2 and −1/2. Both conditions are about slope alone, not intercepts.
Why negative reciprocal
Rotating a line 90° turns a rise of a and run of b into a rise of b and run of −a, so the slope goes from a/b to −b/a. The rule follows from what a quarter turn does to the coordinates.
The horizontal and vertical case
A horizontal line and a vertical line are perpendicular, but their slopes are 0 and undefined, so the product rule cannot be applied. This pair must be handled separately, and it is the standard exception.
Writing a parallel or perpendicular line
Take the slope from the condition and the point from the problem, then use point-slope form. The given line supplies only the slope; the point fixes which of the infinitely many such lines you want.
Step 2: Try It Yourself
Tap and try it out.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Dev Find a Perpendicular
Dev needs a line perpendicular to y = 4x + 1.
- Step 1
The slope is 4, which is 4/1.
Step 4: Your Turn
Practice makes it stick.
Parallel
Problem 1 of 2
Parallel to y = 6x − 2. What slope?
Perpendicular
Problem 2 of 2
Perpendicular to y = 2x. What is the product of the two slopes?
Same or At Right Angles
1 of 8
Parallel to y = 3x + 5. What slope?
2 of 8
Perpendicular to a line of slope 5. What is the perpendicular slope, as a decimal?
3 of 8
Slopes 3 and −1/3. Perpendicular? 1 for yes, 0 for no.
4 of 8
Slopes 2 and 2. Parallel? 1 for yes, 0 for no.
5 of 8
Perpendicular to slope −4. What is the perpendicular slope, as a decimal?
6 of 8
Match each relationship to its slope condition.
Tap a card on the left to start.
7 of 8
Slopes 4 and −4. Perpendicular? 1 for yes, 0 for no.
8 of 8
Perpendicular to slope 1. What slope?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Perpendicular to slope 2. What slope, as a decimal?
What You Learned
- Parallel lines share a slope and differ in intercept.
- Perpendicular slopes are negative reciprocals, so their product is −1.
- Horizontal and vertical lines are perpendicular but escape the product test.