Geometry and measures
Distance and section formulas
Find the distance between two points, the point that divides a segment in a given ratio, and the area of a triangle from its vertices.
Two points on a grid, and you want the distance between them. Go across from one to the other, then up: that is a right-angled triangle with the distance as its hypotenuse. So the distance formula is not a new idea at all — it is Pythagoras' theorem with the two shorter sides worked out by subtracting coordinates.
The distance formula
Between (x₁, y₁) and (x₂, y₂), the distance is √((x₂ - x₁)² + (y₂ - y₁)²). Subtract in whichever order you like — squaring destroys the sign either way.
Worked example
Find the distance between (1, 2) and (4, 6).
Across: 4 - 1 = 3. Up: 6 - 2 = 4.
These are the two shorter sides of the right-angled triangle you have just drawn without drawing it.
The section formula answers a different question: not how far apart two points are, but where a point sits between them. If P divides the segment from A to B in the ratio m : n, then P is m parts of the way along out of m + n. Each coordinate of P is a weighted average — and the weight that goes with A is n, not m, which is the part everyone gets wrong at first.
The section formula
P = ((m·x₂ + n·x₁)/(m + n), (m·y₂ + n·y₁)/(m + n)) for the ratio m : n from (x₁, y₁) to (x₂, y₂). Put m = n = 1 and it collapses to the midpoint, which is the check worth doing whenever you cannot remember which weight goes where.
Try it together
Let us find the point P that divides the segment from (1, 2) to (7, 11) in the ratio 2 : 1.
Two parts of the way from the first point towards the second, so P is nearer (7, 11).
1.The far point (7, 11) is weighted by 2. What is 2 × 7?
Have a go
Have a go on your own: what is the distance between (0, 0) and (9, 12)? Give the answer in units.
Hint: It is the 3, 4, 5 triangle scaled up by three.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.