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MathidiaMathematics Academy
Coordinate geometry and circles
Lesson 12 of 75
Easy practice

Parallel and perpendicular lines

Objective. Use gradient relationships in geometric proofs.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Parallel lines share identical gradients, while perpendicular lines have gradients that multiply to give −1, so a line perpendicular to one with gradient m has gradient −1/m. These facts let you prove geometric properties, such as showing two sides of a shape meet at a right angle by multiplying their gradients. For a line with gradient 2/3, a perpendicular line has gradient −3/2, found by flipping and negating. In coordinate proofs, always state the gradient relationship explicitly before concluding parallelism or perpendicularity.

Worked example

Find the gradient of a line perpendicular to the line 2y=4x−6.

2 … 4
Coordinate grid with the line 2 … 4
  1. 1Rearrange to y=2x−3 to find the original gradient m=2
  2. 2Recall perpendicular gradients satisfy m₁m₂=−1
  3. 3Compute the perpendicular gradient as −1/2
  4. 4State the perpendicular gradient as −1/2

Answer: −1/2

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Common mistake. Students find the perpendicular gradient by just negating, turning gradient 2 into −2 instead of the correct −1/2; perpendicular gradients must be both flipped (reciprocal) and negated.

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