Parallel and Perpendicular Lines
Two lines on the test are either heading the same direction forever, or crashing into each other at a perfect 90°. Knowing the one slope rule for each lets you answer these questions in seconds.
A line with slope 2. A parallel line keeps slope 2; a perpendicular line would have slope -1/2.
Quick rule for choosing the target slope.
The whole game: find the slope of the given line, then keep it (parallel) or flip-and-negate it (perpendicular). Once you have the target slope, plug in a point if they ask for the full equation.
What is the distance between (0, 0) and (3, 4)?
Worked examples
Line k is given by y = -4x + 9. Which of the following lines is parallel to line k?
Line m has slope 2/3. Line n is perpendicular to line m. What is the slope of line n?
Line p passes through the points (2, 1) and (6, 9). Line q is perpendicular to line p and passes through the point (0, 5). What is the equation of line q?
Common pitfalls
Perpendicular requires BOTH steps. The negative reciprocal of 3/4 is -4/3, not -3/4 (only sign changed) or 4/3 (only flipped). Always do both, then verify the product is -1.
Test writers love offering a choice with the same b as the original line. Parallel/perpendicular depends ONLY on the slope m. Ignore the intercept until they ask for a full equation.
If a line is given as Ax + By = C, you can't just grab a number off the front — solve for y first, or use slope = -A/B. Reading the coefficient of x in standard form gives the wrong slope.
A horizontal line has slope 0; a vertical line has undefined slope. You can't take the negative reciprocal of 0, so memorize that horizontal ⊥ vertical directly instead of forcing the formula.
Key takeaways
Parallel lines have equal slopes; the y-intercepts can differ.
Perpendicular slopes are negative reciprocals — flip the fraction AND change the sign.
Two slopes are perpendicular exactly when their product is -1.
Find slope from two points with (y₂ - y₁)/(x₂ - x₁); from standard form Ax + By = C use slope = -A/B.
Horizontal lines have slope 0, vertical lines have undefined slope, and they are perpendicular to each other.
Further reading
Checkpoint
10 questions on Coordinate Geometry, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.