Arc Length Calculator
Calculate arc length, sector area, chord length, and arc-to-chord ratio from radius and central angle. Includes preset common angles, a properties table, and an arc diagram.
Endpoint Calculator
Find a missing endpoint given a midpoint and the other endpoint in 2D or 3D. Distance verification, calculation steps, presets for common problems, coordinate visualization, and formulas.
Endpoint Calculator
Endpoint B — x⧉
7.0000
2(4) − 1 = 7.0000
Endpoint B — y⧉
7.0000
2(5) − 3 = 7.0000
Distance A to B⧉
7.2111
Total segment length
Distance A to M⧉
3.6056
Should be half of AB
Distance B to M⧉
3.6056
Should equal AM: ✓
Midpoint Verified?⧉
✓ Yes
Recomputed midpoint matches given M
Coordinate Plot
Blue = A (given), Green = M (midpoint), Red = B (found)
Distance Breakdown
A to M3.6056
M to B3.6056
A to B (total)7.2111
Calculation Steps
| Step | Calculation |
|---|---|
| Given endpoint A | (1, 3) |
| Given midpoint M | (4, 5) |
| Bx = 2Mx − Ax | 2(4) − 1 = 7.0000 |
| By = 2My − Ay | 2(5) − 3 = 7.0000 |
| Endpoint B | (7.0000, 7.0000) |
| Verify midpoint | (4.0000, 5.0000) = (4, 5) ✓ |
Planning notes, formulas, and examples
About the Endpoint Calculator
If you know a midpoint M and one endpoint A of a line segment, you can find the other endpoint B using the midpoint formula in reverse: B = 2M − A. In 2D, this means Bx = 2Mx − Ax and By = 2My − Ay. In 3D, the same logic extends with a z-coordinate. This is a common coordinate geometry problem in algebra, geometry, and physics courses. For example, if A = (1, 3) and M = (4, 5), then B = (2×4 − 1, 2×5 − 3) = (7, 7). This calculator handles both 2D and 3D cases, shows the step-by-step calculation, verifies the result by recomputing the midpoint and the distances from each endpoint to the midpoint (which must be equal), and plots the points on a coordinate grid. Presets include integer, fractional, and negative coordinate examples, plus 3D cases.
When This Page Helps
Finding a missing endpoint from a midpoint is a standard coordinate geometry problem that students solve frequently in algebra and geometry courses. It gives instant answers with step-by-step reasoning (showing B = 2M − A applied to each coordinate), automatic midpoint verification, distance computation, and a coordinate grid visual — making it an ideal homework checker and teaching aid for 2D and 3D problems.
How to Use the Inputs
- Select 2D or 3D mode for the coordinate system.
- Enter the known endpoint A coordinates (x, y, and optionally z).
- Enter the known midpoint M coordinates.
- Click a preset like "A(1,3) M(4,5)" to load a common geometry problem.
- Review the computed endpoint B, segment length, and half-lengths in the output cards.
- Examine the step-by-step calculation showing Bx = 2Mx − Ax for each coordinate.
- Verify the result: the calculator confirms that the midpoint of A and B equals M.
Formula used
2D: B = (2Mx − Ax, 2My − Ay)
3D: B = (2Mx − Ax, 2My − Ay, 2Mz − Az)
Verification: midpoint of A and B should equal M
Distance = √((Bx−Ax)² + (By−Ay)²)Example Calculation
Result: Endpoint B = (7, 7)
Bx = 2(4) − 1 = 7, By = 2(5) − 3 = 7. Verification: midpoint of (1,3) and (7,7) = (4,5) ✓.
Tips & Best Practices
- Check that all inputs use the same scale and assumptions before trusting the result.
- Compare the answer with the worked example or a rough estimate to catch entry mistakes.
Deriving the Endpoint Formula
The midpoint formula says M = (A + B)/2. To find B when you know A and M, multiply both sides by 2: 2M = A + B, then subtract A: B = 2M − A. Applying this to each coordinate gives Bx = 2Mx − Ax, By = 2My − Ay, and (in 3D) Bz = 2Mz − Az. This derivation is worth understanding because the same "solve for the unknown" approach applies to weighted averages, centroids, and section formulas.
Section Formula Generalization
The midpoint is a special case of the section formula, which divides a segment in ratio m:n. The point P dividing AB in ratio m:n is P = (nA + mB)/(m+n). For the midpoint, m = n = 1, giving P = (A + B)/2. If you know P and A and the ratio, you can solve for B using the same algebraic approach. This extends the endpoint formula to any division ratio, useful in computer graphics for interpolation and in physics for center-of-mass calculations.
Applications in Coordinate Geometry
Endpoint problems appear in many geometric contexts: finding the other vertex of a parallelogram given the midpoint of a diagonal, reflecting a point across a line (the line is the perpendicular bisector), and computing symmetric points. In navigation, if you know your starting position and the midpoint of a route, you can calculate the destination using B = 2M − A with latitude and longitude coordinates (for small distances where curvature is negligible).
Sources & Methodology
Last updated:
Frequently Asked Questions
Use B = 2M − A. Each coordinate of the missing endpoint is twice the midpoint coordinate minus the known endpoint coordinate.
The midpoint formula says M = (A + B)/2. Solving for B gives B = 2M − A.
Yes — apply the same formula to each coordinate: Bx = 2Mx − Ax, By = 2My − Ay, Bz = 2Mz − Az.
Compute the midpoint of A and B: ((Ax+Bx)/2, (Ay+By)/2). It should equal your given midpoint M.
Use the distance formula: d = √((Bx−Ax)² + (By −Ay)²). Each endpoint is exactly d/2 from the midpoint.
Absolutely. The formula works for all real numbers.
More in this topic
Associative Property Calculator
Demonstrate and verify the associative property for addition and multiplication. See counterexamples for subtraction and division with custom numbers, presets, and grouping visuals.
Average Rate of Change Calculator
Calculate the average rate of change (f(b)−f(a))/(b−a) for common functions. Includes secant line slope, comparison with instantaneous rate, interval table, and visual diagram.