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Parallel and perpendicular lines

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Lesson notes

Parallel lines

  • Distinct non-vertical lines with equal gradients and different intercepts are parallel. Vertical lines are parallel to one another but have undefined gradient.
  • y = 3x + 1 and y = 3x - 4 are parallel (both gradient 3).
  • To find the line parallel to y = 2x + 5 through (1, 9): use m = 2, then 9 = 2(1) + c gives c = 7, so y = 2x + 7.

Same gradient, different intercepts

Same gradient, different interceptsO−10−8−6−4−22468−2−112xyy = 3x + 1y = 3x − 4

Perpendicular lines

  • For two non-vertical lines meeting at a right angle, m₁m₂ = −1. Horizontal and vertical lines are also perpendicular, but the product rule cannot use an undefined gradient.
  • The perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
  • A line with gradient 2 has a perpendicular gradient -\frac{1}{2}; gradient -\frac{3}{4} gives \frac{4}{3}.

Negative reciprocal gradients

Negative reciprocal gradientsO−4−3−2−11234−4−3−2−11234xyy = 2xy = −½x

Rearranging first

  • Always rearrange to y = mx + c before reading the gradient.
  • 2y = 6x + 4 becomes y = 3x + 2, so m = 3. x + 2y = 8 becomes y = -\frac{1}{2}x + 4, so m = -\frac{1}{2}.

Finding the equation of a parallel or perpendicular line

  • Step 1: find the required gradient. Step 2: substitute the given point into y = mx + c to find c.
  • The line perpendicular to y = 2x + 1 through (4, 3): m = -\frac{1}{2}; 3 = -\frac{1}{2}(4) + c gives c = 5, so y = -\frac{1}{2}x + 5.

Special cases

  • Horizontal lines (y = k) are perpendicular to vertical lines (x = k); their gradients are 0 and undefined.
  • Do not take the reciprocal of zero: a line perpendicular to a horizontal line is vertical, with equation x = k.

Slides

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Practice questions

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  1. 1.What is true of the gradients of distinct non-vertical parallel lines?

    Easy
    • AThey add to zero
    • BThey are equal
    • CThey multiply to -1
    • DThey differ by 1

  2. 2.Which line is parallel to y = 5x + 3?

    Easy
    • Ay = 3x + 5
    • By = 5x - 7
    • Cy = -5x + 3
    • Dy = \frac{x}{5}

  3. 3.What is the gradient of a line parallel to y = -3x + 1?

    Easy
    • A3
    • B-3
    • C\frac{1}{3}
    • D-\frac{1}{3}

  4. 4.What is the gradient of a line perpendicular to y = 4x?

    Easy
    • A4
    • B-4
    • C\frac{1}{4}
    • D-\frac{1}{4}

  5. 5.What is the gradient of a line perpendicular to y = -\frac{2}{5}x?

    Easy
    • A\frac{2}{5}
    • B-\frac{5}{2}
    • C\frac{5}{2}
    • D-\frac{2}{5}

  6. 6.Which pair of lines is perpendicular?

    Easy
    • Ay = 3x and y = 3x + 1
    • By = 3x and y = -\frac{1}{3}x
    • Cy = 3x and y = -3x
    • Dy = 3x and y = \frac{1}{3}x

  7. 7.What is the gradient of y = 6 - 2x?

    Easy
    • A6
    • B-2
    • C2
    • D-6

  8. 8.What is the gradient of 2y = 8x - 6?

    Medium
    • A8
    • B4
    • C2
    • D-3

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