The formulas, for A(x₁, y₁) and B(x₂, y₂)
A standard method from Coordinates: the steps and worked examples.
Steps
| Find | Formula | Why |
|---|---|---|
| Distance AB | √[(x₂ − x₁)² + (y₂ − y₁)²] | Pythagoras on the horizontal and vertical gaps |
| Mid-point | ((x₁ + x₂)/2, (y₁ + y₂)/2) | the average of the ends |
| Slope | (y₂ − y₁)/(x₂ − x₁) | rise ÷ run |
| Point dividing AB in r : s (NF) | ((s·x₁ + r·x₂)/(r + s), (s·y₁ + r·y₂)/(r + s)) | weighted average, closer to A when r is small |
- Label the points A(x₁, y₁) and B(x₂, y₂).
- Write the formula, then substitute. Put negative coordinates in brackets: 4 − (−2).
- Simplify. Leave a distance as an exact surd unless the question asks for a decimal.
- For a missing coordinate, substitute the known slope or distance and solve the equation.
Worked examples
Example 1: Use the formulas for A(−2, 1) and B(4, 9): distance, mid-point and slope.
AB = √[(4 − (−2))² + (9 − 1)²] = √(36 + 64) = 10
mid-pt = ((−2 + 4)/2, (1 + 9)/2) = (1, 5)
slope = (9 − 1)/(4 − (−2)) = 8/6 = 4/3
a line ⊥ AB has slope −3/4 (4/3 × −3/4 = −1)For the whole topic, read Coordinates: key points, common mistakes and how the DSE tests it.