Straight lines, midpoints and perpendicular lines

Gradient, midpoint and distance

• Gradient m = (y₂ − y₁) ÷ (x₂ − x₁). From A(2, 3) to B(6, 11), m = 8 ÷ 4 = 2.
• Midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). The midpoint of AB is (4, 7).
• Distance = √((x₂ − x₁)² + (y₂ − y₁)²). AB = √(4² + 8²) = √80 = 4√5.

Equations of lines

• y = mx + c, where m is the gradient and c is the y-intercept.
• Through a point (x₁, y₁) with gradient m: y − y₁ = m(x − x₁). Through (1, 5) with gradient 2: y − 5 = 2(x − 1), so y = 2x + 3.
• For a line like 2x + 3y = 12, rearrange to y = −(2/3)x + 4 to read off the gradient (−2/3) and the y-intercept (4).

Parallel and perpendicular lines

• Parallel lines have equal gradients.
• Perpendicular lines have gradients that multiply to −1: m₁ × m₂ = −1. The perpendicular gradient to 3 is −1/3; to −2/5 it is 5/2.
• The perpendicular bisector of AB passes through the midpoint of AB at right angles. For A(2, 3) and B(6, 11): midpoint (4, 7), gradient −1/2, so y − 7 = −(1/2)(x − 4), giving y = −(1/2)x + 9.

Dividing a line in a ratio

• To find P dividing AB in the ratio 1 : 2, go 1/3 of the way from A to B. For A(1, 2) and B(7, 11): P = (1 + 6/3, 2 + 9/3) = (3, 5).

Key terms

Gradient

The steepness of a line: change in y divided by change in x.

y-intercept

Where a line crosses the y-axis.

Midpoint

The point halfway between two points.

Parallel

Lines with equal gradients that never meet.

Perpendicular

Lines at right angles, whose gradients multiply to −1.

Perpendicular bisector

A line through the midpoint of a segment at right angles to it.