The perpendicular slope of any line with slope m is always -1/m, the negative reciprocal
The perpendicular slope of 5 is -1/5
Here's how it works: flip the original slope (5 becomes 1/5), then change the sign. So 5 turns into -1/5. Imagine walking up a steep hill (slope 5)—a perpendicular path would feel like a gentle downhill (slope -1/5).
The perpendicular slope of 4 is -1/4
Same process: flip 4 to 1/4, then flip the sign. Think of rotating a ruler 90 degrees—any steep line becomes a shallow one facing the opposite direction. That's the perpendicular slope for you.
The perpendicular slope of -1 is 1
This one's special: -1 flipped is -1, then the sign flips again to 1. It's the only number that's its own negative reciprocal. Two lines with slopes -1 and 1 will always meet at a perfect right angle.
The perpendicular slope of 1/3 is -3
Treat 1/3 like any fraction: flip it to 3/1, then flip the sign. So 1/3 becomes -3. It's like swapping a gentle ramp for a steep slide in the opposite direction.
Multiply the two slopes; if the product is -1, they're perpendicular
Quick check: slope1 × slope2 = -1 means they're at right angles. No need to graph—just do the math. This works whether you're dealing with fractions or whole numbers.
The perpendicular slope to 3/2 is -2/3
Flip 3/2 to 2/3, then flip the sign to negative. It's like swapping a steep hike for a gentler walk in the opposite direction. This keeps the lines at a perfect 90-degree angle.
No, y = -4 is a horizontal line (slope 0), not perpendicular to any line with slope 1
Horizontal lines never rise—they're flat. A line with slope 1 rises one unit for every one unit it runs, so they're not at right angles. Only a vertical line (undefined slope) is truly perpendicular to y = -4.
The perpendicular slope of 3/4 is -4/3
Flip 3/4 to 4/3, then flip the sign to negative. It's like turning a gentle upward slope into a steeper downward one, ensuring they cross at a right angle.
Start with the opposite reciprocal slope, then plug in a point to solve for b in y = mx + b
Say you know a line passes through (2, 5) and is perpendicular to y = 2x + 1. Its slope is -1/2. Plug into y = -1/2x + b, then solve for b using (2, 5). You'll get the full equation. If you're working with more complex geometry, you might also need to understand how perpendicular bisectors function in different contexts.
An undefined slope belongs to a vertical line where x never changes
Imagine climbing a cliff: your x-coordinate stays the same, but your y-coordinate shoots up. That's an undefined slope—no run, all rise. Honestly, this is the simplest slope to understand once you picture it.
Same slope = parallel; negative reciprocal slope = perpendicular; otherwise = neither
It's a three-way test. Compare slopes directly for parallel, flip and flip the sign for perpendicular, or call it quits if neither applies. No fancy tools needed—just basic arithmetic.
The perpendicular slope of 5/8 is -8/5
Flip 5/8 to 8/5, then flip the sign to negative. It's like turning a gentle upward trend into a steeper downward one, ensuring they cross at a right angle.
The opposite slope of 3 is -3
“Opposite” here means sign flip, not reciprocal flip. It's the simplest slope swap: 3 becomes -3. Think of it as reversing direction on a straight road.
A perpendicular line forms a 90-degree angle with another line
Picture a plus sign (+): the two lines meet at a perfect corner, marked by a tiny square. Or think of a wall meeting the floor—always at right angles. That's perpendicularity in action. If you're curious about other geometric concepts, you might also enjoy learning about perpendicular distances in coordinate geometry.
Known as a jack of all trades and master of none, though he prefers the term "Intellectual Tourist." He spent years dabbling in everything from 18th-century botany to the physics of toast, ensuring he has just enough knowledge to be dangerous at a dinner party but not enough to actually fix your computer.