Parallel and Perpendicular Lines

Points on a plane

Every point on a plane has coordinates (x, y). For two points A(x₁, y₁) and B(x₂, y₂) we can find the midpoint, the distance between them and the gradient of the line joining them.

The midpoint M, the distance AB and the gradient m.

Worked exampleMidpoint, distance and gradient

Find the midpoint, length and gradient of the line joining A(1, 2) and B(7, 10).

Midpoint.
Distance.
Gradient.

Answer: Midpoint (4, 6), length 10, gradient 4/3.

Lines	Condition on gradients
Parallel	m₁ = m₂
Perpendicular	m₁ × m₂ = −1

The lines y = 2x and y = −x/2 have gradients 2 and −1/2: their product is −1, so they cross at a right angle.

Try itA line segment

Choose the coordinates of two points.

Check yourself The midpoint of (2, 4) and (8, 10) is

Check yourself The gradient perpendicular to a line of gradient 3 is

Check yourself The distance from (0, 0) to (3, 4) is

Key terms

Midpoint: The point halfway between two points.
Gradient: The steepness of a line.
Parallel lines: Lines with the same gradient.
Perpendicular lines: Lines whose gradients multiply to −1.

Summary

Midpoint = average of the coordinates.
Distance = √(Δx² + Δy²).
Parallel lines have equal gradients.
Perpendicular gradients multiply to −1.