A worked example
Dividing (1,2) to (9,8) in ratio 3:1 gives P=(7,6.5), three quarters of the way from the first point.
Math calculator
Section Formula Calculator
Find the point dividing a segment internally in a specified ratio. Input changes update both division point and the supporting steps.
Section Formula inputs
Complete the fields
When the result is useful
Section points locate proportional positions in geometry, animation, mapping, and coordinate construction. A related application of Section Formula is equal division.
Reading the calculation
For AP:PB=m:n, point P=((nx₁+mx₂)/(m+n),(ny₁+my₂)/(m+n)). The opposite weights reflect distance from each endpoint. Section Formula also connects with recover an endpoint.
A correct division point therefore depends on choosing the model before entering the numbers.
Working through Section Formula on paper
Form weighted averages with the opposite endpoint weights and divide by m+n.
Details to check in Section Formula
This page uses positive ratio parts for internal division. Reversing m and n moves the point to the complementary location.
When section formula looks surprising, restore the sample and vary second endpoint x₂ by itself.
Checking the model and magnitude
For this section formula calculation, the labels first endpoint x₁, first endpoint y₁, second endpoint x₂, second endpoint y₂, ratio part toward second endpoint m and ratio part toward first endpoint n carry mathematical meaning. A transposed entry can remain numerically valid while describing an entirely different setup.
How the result responds
Increasing m moves the point toward the second endpoint. A small controlled input change is enough to test the expected direction.
The midpoint formula is the special case m=n=1. Checking the requested noun is often enough to select the right model.
Recording the answer
An exact expression can preserve factors, radicals, or π that a decimal hides. Use the representation suited to the next task and label it clearly as division point.
Reproducibility here depends on the inputs more than the interface. Preserve first endpoint x₁, first endpoint y₁, second endpoint x₂, second endpoint y₂, ratio part toward second endpoint m and ratio part toward first endpoint n, the operation shown, and enough unrounded digits for the next calculation.
Preserving the setup for Section Formula
Measure coordinate differences from the first endpoint to the result and to the second endpoint. Their component ratios should agree with m:n.
For Section Formula, copying only the final number discards the assumptions that make it meaningful. Retain the inputs and one independent identity with the answer. This compact record is sufficient for a later review and prevents rounded output from becoming the source for an avoidable second-rounding error.
Questions about Section Formula
Why are the weights opposite?
A point closer to one endpoint receives more weight from that endpoint in the average.
What ratio gives the midpoint?
1:1.
Can ratios be negative?
That describes external division and is outside this page.