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
Try it yourself
5 practice questions on Parallel and Perpendicular Lines, drawn from the question bank. The tutor is one click away if you get stuck.