Midpoint and Section Formula Comparison

Quick answer: Midpoint is the special case of internal division in ratio 1:1. If AP:PB = m:n, then P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). The weight m multiplies endpoint B.

Worked example

For A(0, 2), B(6, 8), find the midpoint and the point dividing AB in ratio 1:2.

1. Midpoint = ((0 + 6)/2, (2 + 8)/2) = (3, 5).
2. For m = 1, n = 2, x = (1×6 + 2×0)/3 = 2.
3. y = (1×8 + 2×2)/3 = 4.
4. The required internal division point is (2, 4).

Check the result

P is one-third of the way from A to B, so AP:PB = 1:2.

Common mistake to avoid

Do not use external-section subtraction for an internal ratio. A point closer to A should receive more weight from A's coordinates.

Try it yourself

1. Find midpoint of (−2, 4) and (4, 10). 2. Divide (0, 0) to (9, 0) in ratio 2:1.

Answers and reasoning

1. (1, 7). 2. (6, 0), since two of the three equal parts lie before P.

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