Key takeaway: The midpoint formula finds the point exactly halfway between two points by averaging their x-coordinates and averaging their y-coordinates, giving the center of the segment that joins them.
Grade 10 – GeometryG-GPE.B.6Geometry and Measurement
Definition of Midpoint Formula
The midpoint formula turns two endpoints on a coordinate grid into the one point that sits halfway between them. Since the midpoint is equally far from both ends, each of its coordinates is the mean of the matching coordinates of the endpoints.
It works in every quadrant and for any tilt, including flat and upright segments. It is also the fastest route to a missing endpoint: knowing one end and the midpoint, you can work backward.
Formula and notation
Let the endpoints be \((p, q)\) and \((r, s)\). The midpoint \(M\) is:
\[ M = ( \frac{p + r}{2},\ \frac{q + s}{2} ) \]
Add the first coordinates and halve the total; then do the same with the second coordinates. The answer is an ordered pair, so the midpoint formula always gives two numbers.
Key facts
- Swapping which endpoint is called first changes nothing, because addition is commutative.
- Each half of the segment has the same length, since the midpoint is equally far from both ends.
- Negative coordinates are added with their signs before dividing by 2.
- Given one endpoint and the midpoint, each missing coordinate equals 2 times the midpoint minus the known one.
Midpoint Formula example
Segment \(\overline{AB}\) has endpoints \(A(-6, 9)\) and \(B(10, -3)\). Find its midpoint, then find the other endpoint of a segment that starts at \(C(1, 4)\) and has the same midpoint.
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\(x\)-coordinate: \(\frac{-6 + 10}{2} = \frac{4}{2} = 2\).
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\(y\)-coordinate: \(\frac{9 + (-3)}{2} = \frac{6}{2} = 3\), so \(M = (2, 3)\).
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For the other endpoint \(D\): \(x_D = 2(2) – 1 = 3\) and \(y_D = 2(3) – 4 = 2\).
The midpoint is \((2, 3)\), and the other endpoint is \(D(3, 2)\); check \(\frac{1+3}{2} = 2\) and \(\frac{4+2}{2} = 3\).
Common mistake
Watch out
For \(A(-6, 9)\) and \(B(10, -3)\), Kai adds the coordinates and reports the midpoint as \((4, 6)\).
Why it is wrong
He forgot to divide by 2. Adding gives \((-6 + 10,\ 9 – 3) = (4, 6)\), which is twice the midpoint. Dividing each sum by 2 gives \((2, 3)\). A quick check: the midpoint’s coordinates must lie between the endpoints’ coordinates.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement