Parallel and perpendicular lines

Parallel and perpendicular

• Parallel lines have the same gradient.
• Perpendicular lines have gradients that multiply to -1: m_1 x m_2 = -1. So the perpendicular gradient is the negative reciprocal: 2 becomes -1/2, and -3/4 becomes 4/3.

Length and midpoint

For A(x_1, y_1) and B(x_2, y_2):

• length AB = sqrt((x_2 - x_1)² + (y_2 - y_1)²)
• midpoint = ((x_1 + x_2)/2, (y_1 + y_2)/2)

Perpendicular bisector

The perpendicular bisector of AB passes through the midpoint of AB and is perpendicular to AB.

Worked example

Find the perpendicular bisector of A(1, 2) and B(5, 10).

1. Midpoint (3, 6)
2. Gradient of AB = 8/4 = 2, so the perpendicular gradient is -1/2
3. y - 6 = -1/2 (x - 3)
4. Multiply by 2: 2y - 12 = -x + 3

Answer: x + 2y - 15 = 0

Check yourself

1. Find the gradient of a line perpendicular to y = 4x - 3. Give your answer as a fraction in its simplest form.
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   -1/4. The negative reciprocal of 4 is -1/4.

2. Find the gradient of a line parallel to 2x - 5y + 1 = 0.
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   0.4. 5y = 2x + 1, so y = 0.4x + 0.2. Parallel lines share the gradient 0.4.

3. Which pair of lines is perpendicular?
   • y = 3x + 1 and y = -3x + 2
   • y = 3x + 1 and x + 3y = 6
   • y = 2x and y = 2x - 5
   • y = 0.5x and y = 2x
   Show answer

   y = 3x + 1 and x + 3y = 6. x + 3y = 6 has gradient -1/3, and 3 x (-1/3) = -1.

4. Find the distance between (1, -2) and (7, 6).
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   10. sqrt(6² + 8²) = sqrt(100) = 10.

5. Find the x-coordinate of the midpoint of (-3, 5) and (7, 1).
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   2. (-3 + 7)/2 = 2.

6. Line L passes through (4, 1) and is perpendicular to y = 2x + 3. Find the y-intercept of L.
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   3. L has gradient -1/2. y - 1 = -1/2 (x - 4) gives y = -x/2 + 3, so the y-intercept is 3.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.