Section Formula

Q: What is the section formula?

The section formula finds the coordinates of a point P that divides a line segment AB in a ratio m:n. For internal division: P = ((m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n)). The midpoint formula is the special case m = n = 1.

Q: What is internal vs external division?

Internal division places P between A and B (m and n both positive). External division places P outside the segment, on the extension of AB. For external, the formula uses subtraction: P = ((m·x₂ − n·x₁)/(m−n), …).

Q: How do I find the centroid of a triangle?

The centroid G is the average of the three vertices: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). It divides each median in a 2:1 ratio measured from the vertex, which the section formula confirms.

Q: Can the section formula find the ratio given the point?

Yes — rearrange to k = AP/PB = (x − x₁)/(x₂ − x). The same formula works for the y-coordinate; both should give the same k if P actually lies on segment AB.

Q: Does it work in 3D?

Yes — just add the z-coordinate using the same logic: P_z = (m·z₂ + n·z₁)/(m+n). Everything else identical.

The section formula tells you the coordinates of a point that divides a line segment AB in a specific ratio m:n. Internal division places the point between A and B; external division places it outside the segment on the extension of AB. The midpoint formula is just the special case m = n = 1.

The Formulas

Name	Formula	Notes
Internal Division (2D)	`P = ((m·x₂ + n·x₁) / (m + n), (m·y₂ + n·y₁) / (m + n))`	P divides A(x₁, y₁) → B(x₂, y₂) internally in ratio m:n. P lies between A and B.
External Division (2D)	`P = ((m·x₂ − n·x₁) / (m − n), (m·y₂ − n·y₁) / (m − n))`	P lies on extension of AB beyond B (or beyond A if m < n). m ≠ n.
Midpoint Formula	`M = ((x₁ + x₂)/2, (y₁ + y₂)/2)`	Special case m = n = 1 of the internal section formula.
Centroid of Triangle	`G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)`	The centroid divides each median in 2:1 ratio. Average of the three vertices.
Section Formula (3D)	`P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n))`	Same logic in three dimensions — add the z-coordinate.
Ratio from Coordinates	`k = AP / PB = (x − x₁) / (x₂ − x)`	Inverse: given the dividing point, find the ratio. Same formula works for y.

Worked Examples

Example 1: Internal division of A(2, 3) and B(8, 9) in ratio 2:1

x = (2·8 + 1·2) / (2 + 1) = 18 / 3 = 6
y = (2·9 + 1·3) / (2 + 1) = 21 / 3 = 7
P = (6, 7)

Example 2: External division of A(1, 2) and B(4, 8) in ratio 3:1

x = (3·4 − 1·1) / (3 − 1) = 11 / 2 = 5.5
y = (3·8 − 1·2) / (3 − 1) = 22 / 2 = 11
P = (5.5, 11) — on AB extended beyond B

Example 3: Midpoint of A(−2, 4) and B(6, −2)

M = ((−2 + 6)/2, (4 + (−2))/2)
M = (4/2, 2/2) = (2, 1)

Frequently Asked Questions

What is the section formula?

The section formula finds the coordinates of a point P that divides a line segment AB in a ratio m:n. For internal division: P = ((m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n)). The midpoint formula is the special case m = n = 1.

What is internal vs external division?