Parallel and perpendicular lines are two of the most important relationships in coordinate geometry. Parallel lines run in the same direction and never meet, while perpendicular lines intersect at a right angle. These relationships are easy to identify when you understand slope.
They matter because they help you compare graphs, build equations, and solve geometry problems on a coordinate plane.
The slope of a line measures its steepness as rise over run. Parallel lines have equal slopes, but different y-intercepts if they are distinct lines. Perpendicular lines have slopes that are negative reciprocals, which means their product is -1 when both slopes are defined.
To write equations of parallel or perpendicular lines, start with the slope relationship, then use a point and a line equation form such as y = mx + b or y - y1 = m(x - x1).
Key Facts
- Slope formula: m = (y2 - y1) / (x2 - x1)
- Slope-intercept form: y = mx + b
- Point-slope form: y - y1 = m(x - x1)
- Parallel lines have the same slope: m1 = m2
- Perpendicular lines have negative reciprocal slopes: m2 = -1 / m1
- For nonzero slopes, perpendicular lines satisfy m1m2 = -1
Vocabulary
- Parallel lines
- Parallel lines are lines in the same plane that never intersect and have equal slopes if they are not vertical.
- Perpendicular lines
- Perpendicular lines are lines that intersect to form a 90 degree angle.
- Slope
- Slope is the ratio of vertical change to horizontal change between two points on a line.
- Negative reciprocal
- A negative reciprocal is found by flipping a nonzero number and changing its sign, such as 3/4 becoming -4/3.
- Y-intercept
- The y-intercept is the point where a line crosses the y-axis, represented by b in y = mx + b.
Common Mistakes to Avoid
- Using opposite signs only for perpendicular slopes is wrong because perpendicular slopes must be negative reciprocals, not just opposites. For example, 2 and -2 are not perpendicular slopes.
- Forgetting that parallel lines need different intercepts is wrong when describing distinct parallel lines. The equations y = 3x + 1 and y = 3x + 1 represent the same line, not two different parallel lines.
- Applying m2 = -1 / m1 to a horizontal line is wrong because a horizontal line has slope 0 and its perpendicular line is vertical with undefined slope. The negative reciprocal rule only works for nonzero defined slopes.
- Mixing up rise and run in the slope formula is wrong because slope is vertical change divided by horizontal change. Reversing the order gives the reciprocal and changes the line relationship.
Practice Questions
- 1 Find the slope of the line through A(2, 5) and B(8, 17). Then write the slope of a line parallel to it and the slope of a line perpendicular to it.
- 2 Write the equation of the line parallel to y = -2x + 7 that passes through the point (3, -4).
- 3 A line has equation y = 3/5x - 2, and another line passes through two points so that its slope is -5/3. Explain whether the two lines are parallel, perpendicular, or neither, and justify your answer.