➕ Algebra 1 · Linear Functions
Parallel and Perpendicular Lines: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Lines on a rectangular court
On a rectangular court, the two sidelines never meet, and each end line meets them in a square corner. Put the court on a graph with the sidelines along y = 0 and y = 5 and the end lines along x = 0 and x = 8 (made-up units). The two sidelines have the same slope, 0, and different y-intercepts, so they are parallel. The two end lines are both vertical, so they are parallel too. Each end line is vertical and each sideline is horizontal, so they are perpendicular.
How it connects: Horizontal with vertical is the special perpendicular pair, where the flip-and-change-sign rule is not used (0 has no reciprocal). The limit: turn the court on the graph and the sidelines would share some slope m while the end lines would have slope −1/m. And the right angle only looks square if both axes use the same scale.
The chess bishop
In chess, a bishop moves along diagonals. Put the board on a graph with square cells. One kind of diagonal goes up and to the right, 1 up for every 1 right, so its slope is 1. The other kind goes up and to the left, so its slope is −1. Two different diagonals of the same kind are parallel, like y = x and y = x + 2. A diagonal of one kind crosses a diagonal of the other kind at a right angle.
How it connects: Slopes 1 and −1 multiply to −1: the reciprocal of 1 is 1, and changing the sign gives −1. The limit: this pair is the easy case. For other slopes, like 3/4, you really need both the flip and the sign change. And it works because the board squares are equal, so the graph has the same scale on both axes.
Walking at the same pace
Ava and Ben walk along the same straight road, in the same direction, at the same steady speed: 80 meters per minute. When they start, Ava is already 200 meters farther along the road than Ben. Let t be the minutes since they started and y the position in meters, measured from Ben’s starting point. Then Ben is at y = 80t and Ava is at y = 80t + 200. The gap stays 200 meters, and Ben never catches up.
How it connects: The same slope (the same speed) and different y-intercepts (different starting spots) give parallel lines, and parallel lines never meet. The limit: if Ben speeds up or Ava slows down, the slopes are different and the lines cross, so Ben can catch up. Also, the right angle between lines would mean nothing here, because time and distance use different units.
Opposites attract
The saying “opposites attract” means that people who are very different from each other are often drawn together. A perpendicular slope is an “opposite” in two ways at once: the sign is opposite, and the fraction is flipped. The slope 5/9 has the perpendicular partner −9/5. Careful: in math, “opposites” alone means the same number with the other sign, like 5/9 and −5/9, and those slopes are not perpendicular.
How it connects: You need both opposites. Changing only the sign (5/9 and −5/9) does not give perpendicular lines, and neither does flipping only (5/9 and 9/5). The product must be exactly −1: (5/9)(−9/5) = −1. The limit: the saying is about people and is not a rule of nature. For slopes, the rule is exact.
The two lines that never met
Why did the two parallel lines never become friends? They had so much in common, but they never met.
How it connects: Same slope and different y-intercepts: the lines never meet. The limit: “so much in common” is only the slope. The y-intercepts are different. If they matched too, the two equations would describe the same line, and the lines would meet everywhere.
Memory help
Parallel has two parallel lines in it
Look at the word: the two ls in the middle of “parallel” are themselves two parallel lines side by side. Parallel lines have the same slope m and different y-intercepts b.
SLIP
To write a parallel or perpendicular line through a point: Slope of the given line. Look at what you need (the same slope for parallel, flip and change sign for perpendicular). Insert the point into y = mx + b. Pull out b and write the equation.
The slope rhyme
“Same m but a different b: parallel lines, you see. Flip the slope and change its sign: perpendicular, you’re doing fine.”
Tilt the plus sign
Draw a plus sign: one horizontal line and one vertical line, the basic perpendicular pair. Now imagine tilting the whole plus sign. Both slopes change, but the corner stays square, and as long as neither line is horizontal or vertical, the two slopes multiply to −1.
Opposite is not enough
Slopes 4 and −4 are opposites, but 4 × (−4) = −16, so the lines are not perpendicular. Slopes 4 and 1/4 are reciprocals, but 4 × (1/4) = 1, so they are not perpendicular either. You need the flip and the sign change: 4 and −1/4.
Solve for y before you read the slope
In 3x + 4y = 8 the number in front of x is 3, but the slope is −3/4. Solve for y first: y = −(3/4)x + 2.