Slope of a line

Every straight line leans by some amount. The inclination θ is the angle the line makes with the positive x-axis, measured anticlockwise. It is always 0° or more and less than 180°.

The slope (or gradient) is m = tan θ. It measures steepness, like the climb of a road.

• θ = 60°: m = tan 60° = √3. θ = 0°: m = 0. θ = 135°: m = tan(180° – 45°) = –1.
• Slope positive: θ is acute and the line rises to the right. Slope negative: θ is obtuse and the line falls to the right.
• A line parallel to the x-axis has slope 0. A line parallel to the y-axis (vertical) has θ = 90° and its slope is not defined.

Slope from two points. Through (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁)/(x₂ – x₁), 'rise over run'. Through (1, 2) and (5, 8): m = 6/4 = 3/2. Either point may go first, as long as you keep the same order on top and bottom.

Parallel and perpendicular.

• Parallel lines have equal slopes, and equal slopes mean parallel.
• Perpendicular lines (neither vertical) have m₁ × m₂ = –1. So to get a perpendicular slope, flip the fraction and change the sign: 3/2 becomes –2/3.

Angle between two lines. With slopes m₁ and m₂, the angle θ between them satisfies tan θ = |(m₂ – m₁)/(1 + m₁m₂)| for the acute angle; the other angle is 180° – θ.

• Slopes 2 and 1/3: (1/3 – 2)/(1 + 2/3) = (–5/3)/(5/3) = –1, so the acute angle is 45° and the other is 135°.
• With θ = 0 the formula gives m₁ = m₂ (parallel), and as θ approaches 90° the bottom goes to 0, giving m₁m₂ = –1.

Slopes answer many old questions quickly:

• Find a value. If the line through (2, 1) and (x, 4) has inclination 45°, its slope is tan 45° = 1, so 3/(x – 2) = 1 and x = 5.
• Collinear points. (1, 1), (3, 5), (4, 7): slopes 4/2 = 2 and 2/1 = 2 are equal and share the point (3, 5), so the three points are on one line.
• Right angle without Pythagoras. A(0, 0), B(4, 2), C(3, 4): slope AB = 1/2, slope BC = 2/(–1) = –2, product –1, so angle B = 90°.
• Parallelogram. Show both pairs of opposite sides have equal slopes.