📏 Parallel and Perpendicular Line Calculator
Parallel lines never meet and have exactly the same slope. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals — flip the fraction and change the sign, so m becomes −1/m. These two relationships solve a whole family of coordinate-geometry problems.
Enter the slope and intercept of any line, plus the point your new lines must pass through, and this calculator writes both new equations and draws all three lines so you can see the relationships.
How to use this calculator
Parallel: keep the slope, solve for a new intercept using your point. Perpendicular: take the negative reciprocal of the slope, then do the same.
- Enter the given line as its slope m and y-intercept b (from y = mx + b).
- Enter the point (x, y) both new lines must pass through.
- Press Calculate.
- Read the parallel and perpendicular equations, and verify them on the graph — same direction, or a perfect right angle.
Parallel and Perpendicular Line Worked Example
Given the line y = 2x + 1, find the lines through (3, 4) parallel and perpendicular to it. Parallel keeps slope 2: with the point, b = 4 − 2·3 = −2, so the parallel line is y = 2x − 2. Perpendicular takes the negative reciprocal: slope −1/2, and b = 4 + (1/2)·3 = 5.5, giving y = −1/2x + 11/2. Check the slopes: 2 × (−1/2) = −1, the signature of perpendicular lines. These are the defaults above.
Frequently asked questions
How do you know if two lines are parallel or perpendicular?
Write both in slope-intercept form. Equal slopes (and different intercepts) means parallel. Slopes whose product is −1 means perpendicular. If neither holds, the lines simply cross at some other angle.
What is a negative reciprocal?
Flip the number over and change its sign: the negative reciprocal of 2 is −1/2, of −3/4 is 4/3, of 5 is −1/5. Perpendicular slopes are always negative reciprocals of each other — that is what forces the 90° angle.
What is the perpendicular to a horizontal line?
A horizontal line has slope 0, and its negative reciprocal does not exist as a number — the perpendicular is a vertical line. Through the point (x₀, y₀) its equation is x = x₀. This calculator returns exactly that when you enter slope 0.
Can parallel lines ever be the same line?
If the slopes and the intercepts are both equal, the "two" lines are actually one line (coincident) — every point is shared. Distinct parallel lines need equal slopes but different intercepts, which guarantees they never meet.