Find the point dividing a segment internally in a specified ratio. Input changes update both division point and the supporting steps.
Dividing (1,2) to (9,8) in ratio 3:1 gives P=(7,6.5), three quarters of the way from the first point.
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.
Form weighted averages with the opposite endpoint weights and divide by m+n.
This page uses positive ratio parts for internal division. Reversing m and n moves the point to the complementary location.
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.
Section points locate proportional positions in geometry, animation, mapping, and coordinate construction. A related application of Section Formula is equal division.
Preserve First endpoint x₁ when checking the Section Formula output. Keep First endpoint y₁ under the same convention and estimate Division point. A controlled change to Second endpoint x₂ should move the Section Formula Division point in a mathematically consistent direction.
Record the original First endpoint x₁ before changing Section Formula. Keep First endpoint y₁ with that record and attach its own Division point. A later Section Formula trial then remains distinguishable from the first calculation.
Keep the units of First endpoint x₁ beside the Section Formula work. Interpret First endpoint y₁ under the same convention. The label attached to Division point should describe the quantity that the Section Formula question actually requests.
A point closer to one endpoint receives more weight from that endpoint in the average.
1:1.
That describes external division and is outside this page.