Parallel lines have equal slopes. Equal steepness in the same direction means they never meet.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Perpendicular lines
Perpendicular slopes are negative reciprocals: their product is −1. A slope of 3 pairs with −1/3.
Flip and negate
To find a perpendicular slope, turn the fraction upside down and change its sign. Both steps are needed.
The awkward pair
A horizontal line has slope 0 and a vertical line has undefined slope. They are perpendicular, but the product rule cannot show it.
Writing the equation
Take the required slope, use the given point, and apply point-slope form.
This is the integration
Slope came from algebra and perpendicularity from geometry. On a grid they are one idea.
The slope conditions
Parallel lines have equal slopes; perpendicular lines have slopes multiplying to −1, so each is the negative reciprocal of the other. Both conditions concern slope alone.
Why negative reciprocal
Rotating a line a quarter turn turns a rise of a over run b into a rise of b over run −a. The rule is a consequence of what a 90° rotation does to coordinates.
The horizontal-vertical exception
A horizontal and a vertical line are perpendicular, but their slopes are 0 and undefined, so the product rule cannot apply. This pair must be handled separately every time.
Writing such a line
Take the slope from the condition and a point from the problem, then use point-slope form. The given line supplies only the slope; the point selects which of infinitely many parallels 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 Tomas Write a Perpendicular
Tomas needs the line through (2, 5) perpendicular to y = 4x − 1.
- Step 1
The given slope is 4.
Step 4: Your Turn
Practice makes it stick.
The Perpendicular
Problem 1 of 2
A line has slope 4. What is the perpendicular slope, as a decimal?
The Parallel
Problem 2 of 2
A line has slope −3. What slope is parallel to it?
Parallel or Perpendicular
1 of 8
Slope 2. Perpendicular slope, as a decimal?
2 of 8
Slope 5. Parallel slope?
3 of 8
Slope −0.5. Perpendicular slope?
4 of 8
Slopes 3 and −1/3. What is their product?
5 of 8
Slopes 2 and 2. Parallel or perpendicular? 1 parallel, 2 perpendicular.
6 of 8
Slope 1. Perpendicular slope?
7 of 8
Put the steps for writing a perpendicular line in order.
- 1Flip it and change its sign.
- 2Substitute that slope and the given point into point-slope form.
- 3Rearrange into slope-intercept form.
- 4Read the slope of the given line.
8 of 8
Slope 0.25. Perpendicular slope?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A line has slope 3. What is the perpendicular slope, to two decimal places?
Question 2 of 2
What is the product of two perpendicular slopes?
What You Learned
- Parallel lines have equal slopes.
- Perpendicular slopes are negative reciprocals, with product −1.
- Find the required slope first, then use point-slope form.