Midpoint and section formula on a line
Midpoint averages coordinates; a ratio m:n uses a weighted average.
Key formulas
Midpoint
A(2,5), B(8,1) → M(5,3).
Ratio m:n
Dividing AB in the ratio 2:1 means twice as close to B as the simple average — use the weighted formula carefully.
Internal vs external
External division replaces + with − in the formula. Read the question for “internally” or “externally”.
Worked example
Worked example
Problem. Find the midpoint of A(−3, 4) and B(5, −2). Then divide AB internally in the ratio 1 : 2.
Solution. Midpoint M = ((−3+5)/2, (4−2)/2) = (1, 1). Section 1:2: P = (2·A + 1·B)/3 = ((−6,8)+(5,−2))/3 = (−1,6)/3 = (−1/3, 2).
Check
For 1:1 the section formula collapses to the midpoint.
Review quiz
Midpoint of (0,−6) and (8,10) is:
Practice questions
- Midpoint of (−3,4) and (5,−2).
- Divide (1,1) and (7,4) internally in 1:2.
Study notes
Which weight on which point?
The ratio m:n near A:B puts n with A and m with B in the numerator.