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Section Formula (Internal Division)

Coordinate Geometry · Class X

Divide segment AB in ratio m:n internally and read off the dividing point P.

-2
type a value
-1010
-1
type a value
-1010
6
type a value
-1010
5
type a value
-1010
Live values
  • P.x (m:n = 1:1, midpoint)2
  • P.y (m:n = 1:1)2
  • P.x (m:n = 2:1)3.3333
  • P.y (m:n = 2:1)3
  • Length AB10
xy
  • Line through A and B

Formulas in this lab

  • P.x (m:n = 1:1, midpoint)
    (x1+x2)/2(x_1+x_2)/2
  • P.y (m:n = 1:1)
    (y1+y2)/2(y_1+y_2)/2
  • P.x (m:n = 2:1)
    (2x2+x1)/3(2x_2 + x_1)/3
  • P.y (m:n = 2:1)
    (2y2+y1)/3(2y_2 + y_1)/3
  • Length AB
    ∣AB∣|AB|
Tip: Ratio 1:1 always gives the midpoint , section formula generalizes the midpoint formula.

Frequently asked questions

▶What is the section formula?

If point P divides segment from $A(x_1, y_1)$ to $B(x_2, y_2)$ internally in the ratio $m:n$, then $P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right)$. When $m = n$, P is the midpoint of AB.

▶How do I use this lab?

Set A and B with the sliders, then choose ratio m:n. The lab marks P on AB and shows its coordinates. Try $A(-2, 1)$, $B(4, 7)$, ratio 1:2 to see $P(0, 3)$.

▶Common mistake on this topic

Students swap m and n: the m goes with the second point's coordinates and n with the first. Writing the formula carefully before substitution helps. Also, midpoint is a special case of $m = n = 1$, not a different formula.

▶Where do we use the section formula?

Town planners dividing a road between two villages in a fixed ratio, or marking the boundary of a plot between two heirs, use this idea. In computer graphics, smooth lines and curves are drawn by dividing segments in ratios.