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Slope, Parallel Lines, and Perpendicular Lines
Slope and Line Relationships
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Slope describes how steep a line is and how it changes as you move from left to right on a coordinate plane. It is one of the most important ideas in coordinate geometry because it connects graphs, equations, and real world rates of change. Parallel and perpendicular lines can be identified quickly by comparing their slopes. Learning these relationships helps students graph accurately and write equations with confidence.
A line's slope is found by comparing vertical change to horizontal change, often called rise over run. Parallel lines have the same slope, so they point in the same direction and never meet. Perpendicular lines meet at right angles, and their slopes are negative reciprocals of each other when both slopes are defined. These rules make it possible to classify lines, check graphs, and build equations from points or slope information.
Key Facts
- Slope formula: m = (y2 - y1) / (x2 - x1)
- Slope-intercept form:
- Parallel lines have equal slopes:
- Perpendicular lines have slopes that are negative reciprocals: m1 x m2 = -1
- Horizontal lines have slope and equations of the form
- Vertical lines have undefined slope and equations of the form
Vocabulary
- Slope
- Slope is the ratio of vertical change to horizontal change between two points on a line.
- Parallel lines
- Parallel lines are lines in the same plane that never intersect and have the same slope.
- Perpendicular lines
- Perpendicular lines intersect at a right angle and usually have slopes that are negative reciprocals.
- Negative reciprocal
- A negative reciprocal is found by flipping a fraction and changing its sign, such as 2/3 becoming -3/2.
- y-intercept
- The y-intercept is the point where a line crosses the y-axis, written as in .
Common Mistakes to Avoid
- Using the same sign for perpendicular slopes, which is wrong because perpendicular slopes must be negative reciprocals, not just reciprocals. For example, the perpendicular slope of 3 is -1/3, not 1/3.
- Thinking parallel lines must have the same y-intercept, which is wrong because parallel lines only need the same slope. Different y-intercepts place them at different heights.
- Mixing up rise and run order in the slope formula, which is wrong because the subtraction in the numerator and denominator must match the same point order. If you do not keep the order consistent, you get the wrong slope.
- Saying a vertical line has slope 0, which is wrong because slope is rise divided by run and a vertical line has run 0. Division by zero is undefined, so the slope is undefined.
Practice Questions
- 1 Find the slope of the line through the points (2, 5) and (6, 13). Then state whether a line with slope 2 is parallel, perpendicular, or neither to this line.
- 2 Line A has equation y = -3x + 4. Write an equation for a line parallel to Line A through the point (1, -2), and write an equation for a line perpendicular to Line A through the same point.
- 3 Explain why two different horizontal lines are parallel, and explain why a horizontal line and a vertical line are perpendicular.