Find the endpoint of a line segment given one endpoint and the midpoint.
Last updated: April 2026 | By Patchworkr Team
Given one endpoint and the midpoint of a segment, you can find the other endpoint.
This is useful in geometry for finding missing points in bisected segments.
Identify the known endpoint (x₁, y₁)
Why: The endpoint formula requires one starting point. This is your reference coordinate from which the other point is calculated.
Identify the midpoint coordinates (m_x, m_y)
Why: The midpoint is equidistant from both endpoints. It serves as the pivot point that defines the segment's center, allowing us to work backwards to find the missing endpoint.
Calculate x₂ using the formula: x₂ = 2 × m_x - x₁
Why: This rearranges the midpoint formula ((x₁ + x₂)/2 = m_x) to solve for the unknown x-coordinate. Multiplying midpoint by 2 and subtracting x₁ isolates x₂.
Calculate y₂ using the formula: y₂ = 2 × m_y - y₁
Why: The same principle applies to the y-coordinate. Both coordinates follow the same algebraic relationship derived from the midpoint definition.
Verify: Show the resulting endpoint (x₂, y₂)
Why: Always verify by computing the midpoint of (x₁, y₁) and (x₂, y₂). It should equal (m_x, m_y), confirming your calculation is correct.
Scenario
Finding the Other End of a Line Segment
You have a line segment with one known endpoint at (1, 2). Its midpoint is at (4, 5). What are the coordinates of the other endpoint?
Step 1: Collect Given Values
Step 2: Apply X-Coordinate Formula
Step 3: Apply Y-Coordinate Formula
Step 4: Combine Coordinates
Verification: Check Midpoint
Result
Interpretation
The segment connects (1, 2) and (7, 8), with its midpoint at (4, 5). The other endpoint is located 6 units right and 6 units up from the starting point, confirming equal distances to the midpoint on both axes.
Use the midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Yes. Add the z-coordinate: z₂ = 2 × m_z - z₁
Yes, coordinates can be any real number, positive or negative.
Then the other endpoint is at the same location, so the segment has zero length.
Yes, given one endpoint and a midpoint uniquely determines the other endpoint.
Construction, design, navigation, and any geometry problem involving line segments.
Yes, calculate the midpoint of your two endpoints and it should match the given midpoint.
The formulas work with any real numbers, including decimals and fractions.
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