Endpoint Calculator

Find the missing endpoint given the midpoint and one known endpoint, with step-by-step solutions.

Enter Known Values

Known Endpoint A (x1, y1)
Midpoint M (mx, my)

Result

Missing Endpoint B
(8, 11)
coordinates
Endpoint A (2, 3)
Midpoint M (5, 7)
Endpoint B (found) (8, 11)
Distance A to B --
Distance A to M --

Step-by-Step Solution

B = (2mx - x1, 2my - y1)

Understanding the Endpoint Formula

The endpoint formula is derived from the midpoint formula. If M is the midpoint of segment AB, then M = ((x1 + x2) / 2, (y1 + y2) / 2). By rearranging this formula to solve for the unknown endpoint B = (x2, y2), we get x2 = 2mx - x1 and y2 = 2my - y1.

This relationship is fundamental in coordinate geometry. The midpoint divides a line segment into two equal halves, so if you know the midpoint and one endpoint, you can always find the other endpoint by reflecting the known point across the midpoint.

Key Formulas

Midpoint Formula

Find the midpoint M of a line segment with endpoints A and B.

M = ((x1 + x2)/2, (y1 + y2)/2)

Endpoint Formula

Find missing endpoint B given midpoint M and endpoint A.

B = (2mx - x1, 2my - y1)

Distance Formula

Find the distance between two points in a coordinate plane.

d = sqrt((x2-x1)^2 + (y2-y1)^2)

Verification

Verify by checking that M is the midpoint of A and B.

((x1+x2)/2, (y1+y2)/2) = M

How the Endpoint Formula Is Derived

Start with the midpoint formula: mx = (x1 + x2) / 2. Multiply both sides by 2: 2mx = x1 + x2. Subtract x1 from both sides: x2 = 2mx - x1. The same process applies to the y-coordinate: y2 = 2my - y1.

Geometric Interpretation

Geometrically, finding the missing endpoint is equivalent to reflecting point A across the midpoint M. The midpoint M is exactly halfway between A and B, so B is the same distance from M as A is, but in the opposite direction. This reflection property makes the endpoint formula intuitive: you travel from A to M, then continue the same distance and direction beyond M to reach B.

Practical Applications

  • Computer Graphics: Finding symmetric points for rendering shapes, animations, and transformations in 2D and 3D environments.
  • Navigation: Determining a destination point when you know a waypoint and a starting location.
  • Engineering: Locating structural support points when the center and one anchor point are known.
  • Physics: Calculating equilibrium positions and center-of-mass problems in mechanics.
  • Surveying: Finding boundary markers when a reference point and a midpoint are established.

Worked Example

Suppose endpoint A = (1, 4) and midpoint M = (3, 6). Find endpoint B.

  1. Use the formula: x2 = 2(3) - 1 = 5
  2. Use the formula: y2 = 2(6) - 4 = 8
  3. Therefore, B = (5, 8)
  4. Verify: midpoint of (1,4) and (5,8) = ((1+5)/2, (4+8)/2) = (3, 6) = M. Correct!

Common Mistakes to Avoid

  • Confusing the midpoint with an endpoint in the formula.
  • Forgetting to multiply the midpoint coordinates by 2 before subtracting.
  • Mixing up the x and y coordinates when entering values.
  • Not verifying the result by plugging back into the midpoint formula.

Extension to 3D

The endpoint formula extends naturally to three dimensions. If A = (x1, y1, z1) and M = (mx, my, mz), then B = (2mx - x1, 2my - y1, 2mz - z1). The same reflection principle applies in any number of dimensions.