Midpoint and dividing point
The midpoint M of the segment joining A(x₁, y₁) and B(x₂, y₂) is the average of the coordinates. To divide AB internally in the ratio m : n, use the section formula.
Key formula
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Point dividing A→B in ratio m : n:
P = ((nx₁ + mx₂)/(m + n), (ny₁ + my₂)/(m + n))
Reading the ratio
In the ratio m : n, m is the part next to A and n the part next to B; the point P lies nearer the smaller part. If the point divides the segment externally one of the weights becomes negative, but at SPM level most problems use internal division. The midpoint is the special case m : n = 1 : 1.
The section formula can be read as a weighted average: the coordinate nearer B carries the weight m and the coordinate nearer A carries the weight n. When m = n the two weights are equal and the point sits exactly halfway, which is why the midpoint is simply the average of the coordinates. A common exam task reverses the process: given the midpoint or dividing point together with one endpoint, rearrange the formula to find the other endpoint, for example x₂ = 2xM − x₁ for a midpoint. Always sketch the segment first so you can check that an internal dividing point really lies between A and B.
Worked example
Let A(2, 1) and B(8, 7). The midpoint is M = ((2 + 8)/2, (1 + 7)/2) = (5, 4). The point P dividing AB in the ratio 1 : 2 is P = ((2·2 + 1·8)/3, (2·1 + 1·7)/3) = ((4 + 8)/3, (2 + 7)/3) = (4, 3).
Remember
- Match m with x₂, y₂ and n with x₁, y₁ in the formula.
- The denominator is always m + n.
- Midpoint means the ratio 1 : 1.