Parallel and perpendicular lines
Objective. Use gradient relationships to construct line equations.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Parallel lines have equal gradients, while perpendicular lines have gradients that multiply to give −1, so the perpendicular gradient is the negative reciprocal of the original. If a line has gradient 2, any parallel line also has gradient 2, and any perpendicular line has gradient −1/2. To find the equation of a line parallel or perpendicular to y=3x+4 through a given point, first identify the correct gradient (3 for parallel, −1/3 for perpendicular), then use y−y1=m(x−x1) with that point. For example, a line perpendicular to y=3x+4 through (6,1) has gradient −1/3, giving y−1=−1/3(x−6). Always check the product of gradients equals −1 to confirm perpendicularity.
Worked example
Find the equation of the line perpendicular to y = 2x − 3 that passes through (4, 5).
- 1Gradient of given line is 2, so perpendicular gradient is −1/2
- 2Use point (4,5) in y−y1=m(x−x1): y−5=−1/2(x−4)
- 3Expand: y−5=−1/2x+2
- 4Rearrange: y=−1/2x+7
Answer: y = −½x + 7
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Practice
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Start practisingCommon mistake. Pupils use the same gradient for a perpendicular line instead of the negative reciprocal, confusing it with the parallel case. For a line of gradient 4, they might use 4 again instead of −1/4 for the perpendicular line. Always flip and negate the gradient for perpendicular lines.
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