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Section Formula Calculator — Point Dividing a Line Segment in a Ratio

Find the point that divides the segment from A(x₁, y₁) to B(x₂, y₂) internally or externally in the ratio m : n with the section formula, plus the midpoint and the distances AP and PB.

A: x₁

A: y₁

B: x₂

B: y₂

Ratio part m (toward B)

Ratio part n (toward A)

Division

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How it works

Internal: P = ((m·x₂ + n·x₁)/(m + n), (m·y₂ + n·y₁)/(m + n)) · External: P = ((m·x₂ − n·x₁)/(m − n), (m·y₂ − n·y₁)/(m − n)) · midpoint: m = n = 1 (internal)

The section formula gives the coordinates of the point P that splits segment AB so that AP : PB = m : n. For internal division P lies between A and B and each coordinate is a weighted average, with the weight m attached to B and n attached to A. For external division P lies on the line AB extended beyond one endpoint, and the plus signs become minus signs. The midpoint formula is the special case m = n = 1 (NCERT Class 10 Mathematics, Chapter 7, Coordinate Geometry). Results are exact fractions.

Worked example

A(2, 3), B(8, 9), ratio 1 : 2 internally. x = (1·8 + 2·2)/(1 + 2) = 12/3 = 4 and y = (1·9 + 2·3)/3 = 15/3 = 5, so P = (4, 5). Check: AP = √8 = 2.828427 and PB = √32 = 5.656854, and 2.828427 ÷ 5.656854 = 0.5 = 1/2.

Sources: NCERT Class 10 Mathematics — Chapter 7 Coordinate Geometry · Wolfram MathWorld — Line Segment

Frequently asked questions

What is the section formula?

For a point dividing A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : n, P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). Externally, replace both plus signs with minus signs.

Which number goes with which point?

m multiplies the coordinates of B and n multiplies those of A — the weight of each endpoint is the part of the ratio next to the other one. AP : PB = m : n.

How is the midpoint formula related?

The midpoint divides AB in the ratio 1 : 1, so the section formula becomes ((x₁ + x₂)/2, (y₁ + y₂)/2).

What is external division?

P lies on line AB but outside the segment, still with AP : PB = m : n. If m > n it is beyond B; if m < n it is beyond A.

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This calculator is for educational purposes. Double-check important results before acting on them.