Find the point that divides a directed line segment AB in a given ratio m:n. Supports internal and external division with section formula, distance calculations, and visual segment diagrams.
| Formula | x-coordinate | y-coordinate |
|---|---|---|
| Internal Division | (mx₂ + nx₁) / (m + n) | (my₂ + ny₁) / (m + n) |
| External Division | (mx₂ − nx₁) / (m − n) | (my₂ − ny₁) / (m − n) |
| Midpoint (1:1) | (x₁ + x₂) / 2 | (y₁ + y₂) / 2 |
| Point | x | y | Distance from A |
|---|---|---|---|
| A | 1.0000 | 2.0000 | 0.0000 |
| P | 5.0000 | 6.0000 | 5.6569 |
| B | 7.0000 | 8.0000 | 8.4853 |
| Midpoint M | 4.0000 | 5.0000 | 4.2426 |
| Step | Computation | Value |
|---|---|---|
| Given A | (1, 2) | — |
| Given B | (7, 8) | — |
| Ratio m:n | 2:1 | — |
| Pₓ | (2×7 + 1×1) / (2+1) | 5.0000 |
| Pᵧ | (2×8 + 1×2) / (2+1) | 6.0000 |
The Ratios of Directed Line Segments Calculator finds the exact point P that divides a line segment from A to B in a specified ratio m:n. This is a core concept in coordinate geometry and analytic geometry courses, used extensively in problems involving partitioning, midpoints, centroids, and geometric constructions.
The calculator implements the Section Formula for both internal and external division. For internal division, point P lies between A and B, and the formula is P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). For external division, P lies on the line through A and B but outside the segment, using P = ((mx₂ − nx₁)/(m−n), (my₂ − ny₁)/(m−n)).
Enter the coordinates of endpoints A and B along with the desired ratio m:n, and the calculator computes point P, all distances (AB, AP, PB), the midpoint for reference, and verifies the ratio. A color-coded position bar shows exactly where P falls along the segment, making it intuitive to understand how the ratio affects placement. The section formula reference table and step-by-step calculation walkthrough make this an ideal study companion for geometry students and anyone working with directed line segments.
Ratios of Directed Line Segments Calculator helps you solve ratios of directed line segments problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter A: x₁, A: y₁, B: x₂ once and immediately inspect Point P (x, y), Distance AB, Distance AP to validate your work.
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 (special case m=n): M = ((x₁+x₂)/2, (y₁+y₂)/2).Result: Point P (x, y) shown by the calculator
Using the preset "A(1,2) B(7,8) 2:1", the calculator evaluates the ratios of directed line segments setup, applies the selected algebra rules, and reports Point P (x, y) with supporting checks so you can verify each transformation.
This calculator takes A: x₁, A: y₁, B: x₂, B: y₂ and applies the relevant ratios of directed line segments relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Point P (x, y), Distance AB, Distance AP, Distance PB to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
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The section formula gives the coordinates of a point that divides a line segment joining two points in a given ratio. It has two forms: one for internal division (between the points) and one for external division (outside the segment).
Internal division means point P lies between A and B on the segment. External division means P lies on the line through A and B but outside the segment.
When m = n, internal division gives the midpoint of the segment. External division is undefined because the denominator (m − n) becomes zero.
Yes — "directed" means the order of A and B matters. Dividing from A to B in ratio 2:1 places P closer to B, while dividing from B to A in 2:1 places P closer to A.
It appears in finding centroids, medial points, trisection points, and in computer graphics for interpolation between two points.
Negative ratios are equivalent to external division. The calculator provides an explicit external division mode for clarity.
Calculate the distance between two points in 2D or 3D using Euclidean, Manhattan, Chebyshev, and Minkowski metrics. Bearing, direction angle, component breakdown, and distance metrics reference.
Calculate slope (rise/run) from two points. Find angle of inclination, parallel and perpendicular slopes, line equations, distance, and midpoint with an interactive graph.
Simplify ratios to lowest terms using GCD. Handles two-part (a:b) and three-part (a:b:c) ratios with step-by-step simplification, decimal equivalents, and visual comparison bars.