Free Tool · Two Modes · Graph

Midpoint Calculator

Find the midpoint of a segment, or work backwards to find a missing endpoint given the midpoint.

1 Endpoint 1 (x₁, y₁)
2 Endpoint 2 (x₂, y₂)
Preview: fill the coordinates

What this midpoint calculator does

This tool works in two directions. In the first mode, it takes two endpoints of a segment and returns the midpoint — the point exactly halfway between them. In the second mode, it reverses the calculation: given one endpoint and the midpoint, it finds the missing other endpoint. Every result includes a graph showing the two known points (or all three when working backwards) and the segment joining them, along with the full step-by-step working using the midpoint formula.

The midpoint formula

Midpoint of two points
M = ( x1 + x2 2 , y1 + y2 2 )
Finding a missing endpoint
Q = (2mx − x1,   2my − y1)

Where P = (x₁, y₁) is the known endpoint and M = (mx, my) is the midpoint.

Variable definitions

  • (x₁, y₁) — coordinates of the first endpoint.
  • (x₂, y₂) — coordinates of the second endpoint.
  • (mx, my) — coordinates of the midpoint.
  • Q — the missing endpoint in endpoint mode.

Worked examples

Example 1: Midpoint of (2, 3) and (8, 9)

  1. x-coordinate: (2 + 8) / 2 = 10 / 2 = 5.
  2. y-coordinate: (3 + 9) / 2 = 12 / 2 = 6.
  3. Result: M = (5, 6)

Example 2: Midpoint of (−4, 2) and (6, −8)

  1. x-coordinate: (−4 + 6) / 2 = 2 / 2 = 1.
  2. y-coordinate: (2 + (−8)) / 2 = −6 / 2 = −3.
  3. Result: M = (1, −3)

Example 3: Fractional midpoint of (1, 0) and (2, 1)

  1. x-coordinate: (1 + 2) / 2 = 3/2 = 1.5.
  2. y-coordinate: (0 + 1) / 2 = 1/2 = 0.5.
  3. Result: M = (3/2, 1/2)

Example 4: Missing endpoint — one endpoint (2, 3), midpoint (5, 6)

  1. Unknown endpoint x-coordinate: 2 × 5 − 2 = 10 − 2 = 8.
  2. Unknown endpoint y-coordinate: 2 × 6 − 3 = 12 − 3 = 9.
  3. Result: Q = (8, 9)
  4. Check: midpoint of (2, 3) and (8, 9) = (5, 6) ✓

Example 5: Missing endpoint — endpoint (0, 0), midpoint (4, 7)

  1. x-coordinate: 2 × 4 − 0 = 8.
  2. y-coordinate: 2 × 7 − 0 = 14.
  3. Result: Q = (8, 14)
  4. Interpretation: the midpoint is the average of the endpoints, so if one endpoint is the origin, the other must be twice the midpoint.

Where the midpoint appears in Class 9 & 10

The midpoint formula is a small but fundamental tool in coordinate geometry:

  • Perpendicular bisectors — the perpendicular bisector of a segment passes through its midpoint.
  • Circle centre — if you know the endpoints of a diameter, the centre of the circle is their midpoint.
  • Parallelograms — the diagonals of a parallelogram bisect each other; the intersection is the midpoint of both diagonals.
  • Median of a triangle — a median connects a vertex to the midpoint of the opposite side.
  • Real-world application — finding the location halfway between two cities, or the average position of two objects.

Common mistakes to avoid

  • Forgetting to divide by 2. Adding the coordinates is not enough. The midpoint is the average of the coordinates, so you must divide each sum by 2.
  • Mixing up x-coordinates with y-coordinates. Add x's together and divide by 2 to get the midpoint's x. Do the same with y's separately. Do not mix the axes.
  • Sign errors with negative coordinates. For example, (−4 + 6) / 2 = 2/2 = 1, not −5. Write down the addition carefully.
  • Using the average of x and y for a single coordinate. The midpoint is a point (two numbers). It is not a single number obtained by averaging all four input numbers.
  • Missing endpoint formula reversed. To find the missing endpoint, you must subtract the known endpoint from twice the midpoint, not add. Q = 2M − P.
  • Assuming the midpoint lies on the segment's perpendicular bisector somewhere else. The midpoint is on the segment itself — it is the point halfway between the endpoints.

FAQ

What is the midpoint formula?

The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂) / 2, (y₁ + y₂) / 2). It is the average of the two x-coordinates paired with the average of the two y-coordinates.

How do I find a missing endpoint given one endpoint and the midpoint?

Use the midpoint formula in reverse. If M is the midpoint and P is the known endpoint, then the unknown endpoint Q satisfies Q = 2M − P. That is, double each coordinate of the midpoint and subtract the corresponding coordinate of the known endpoint.

Does the midpoint divide the segment in half?

Yes. By definition, the midpoint is the point that divides the segment into two equal lengths. The distance from either endpoint to the midpoint is half the total segment length.

What if the two endpoints are the same point?

If both endpoints are the same, the segment has zero length and the midpoint is that point. This is the degenerate case of a line segment.

Does the order of the endpoints matter?

No. Addition is commutative, so (x₁ + x₂) / 2 = (x₂ + x₁) / 2. The midpoint is the same whether you call one point 'first' or 'second'.

How is the midpoint used in geometry?

The midpoint is used to construct perpendicular bisectors, prove that a point divides a segment equally, find the centre of a circle from a diameter, and identify vertices of a parallelogram (whose diagonals bisect each other at the midpoint).

Related tools

Related Hira Academy resources

The midpoint formula is part of the coordinate geometry chapter in Class 9 and appears in various proofs in Class 10. Revise with our free Class 9 notes and Class 10 notes.

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