Parallel & Perpendicular Lines Calculator
Find parallel and perpendicular line equations through a given point
Original Line (y = mx + b)
Point (x, y)
Visual Representation
Step-by-Step Solution
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines never intersect and have the same slope:
Perpendicular Lines
Perpendicular lines intersect at 90° angles. Their slopes are negative reciprocals:
Real-World Applications
- Architecture: Walls parallel to each other, floors perpendicular to walls
- Road design: Parallel lanes, perpendicular intersections
- Coordinate geometry and proofs
- Computer graphics and game development
- Engineering: Support beams and structures
Frequently Asked Questions
How do I find a parallel line?
Use the same slope as the original line, then use the given point to find the y-intercept.
How do I find a perpendicular line?
Take the negative reciprocal of the original slope (flip the fraction and change the sign), then use the given point to find the y-intercept.
What if the slope is 0?
A horizontal line (slope 0) has a perpendicular vertical line (undefined slope: x = c).
What if the slope is undefined?
A vertical line (undefined slope) has a perpendicular horizontal line (slope 0: y = c).
Can parallel lines have different y-intercepts?
Yes! Parallel lines must have the same slope but different y-intercepts (otherwise they'd be the same line).
How do I verify lines are perpendicular?
Multiply the slopes. If the product equals -1, the lines are perpendicular.
What's a negative reciprocal?
Flip the fraction and change the sign. For example, the negative reciprocal of 2/3 is -3/2, and the negative reciprocal of -4 is 1/4.
Do perpendicular lines always intersect?
Yes, perpendicular lines always intersect at exactly one point, forming a 90° angle.