Midpoint Formula
In short: The midpoint formula averages the x-coordinates and the y-coordinates of two endpoints to give the exact point that splits a segment into two equal halves.
GeometryGeometryG-GPE.B.6
What is Midpoint Formula?
The midpoint formula locates the exact center of a segment drawn on the coordinate plane. Because the center sits halfway across and halfway up, each of its coordinates is simply the average of the matching endpoint coordinates.
For endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the center is \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). It lies on the segment and is the same distance from both ends. Run the idea backward and a known center plus one endpoint reveals the missing endpoint.
How to spot it
- A problem asks for the center, the halfway point or the point that bisects a segment given by coordinates.
- Two endpoints are listed as ordered pairs and the words midpoint, bisect or equidistant appear.
- One endpoint and the center are known, and you must locate the other end.
- A map or grid asks where two travelers should meet so each walks the same distance.
The method
- Write the endpoints as \((x_1, y_1)\) and \((x_2, y_2)\), keeping each pair together.
- Add the two x-values and halve the sum; then add the two y-values and halve that sum.
- Present the answer as an ordered pair \((x_M, y_M)\).
- Check by comparing offsets: the center must be the same horizontal and vertical steps from each endpoint.
Midpoint Formula: solved example
Problem
On a park map, two hikers start at \(A(-3, 1)\) and \(B(9, 7)\) and agree to meet halfway between them. Find the meeting point.
Solution
Average the x-values: \(\frac{-3 + 9}{2} = \frac{6}{2} = 3\). Average the y-values: \(\frac{1 + 7}{2} = \frac{8}{2} = 4\).
Check the offsets: from \(A\) to \((3, 4)\) is 6 right and 3 up, and from \((3, 4)\) to \(B\) is again 6 right and 3 up. The hikers meet at \((3, 4)\).
Try it
Question 1
Find the midpoint of the segment with endpoints \((-8, 5)\) and \((4, -11)\).
Show the answer
\(\left(\frac{-8 + 4}{2}, \frac{5 + (-11)}{2}\right) = (-2, -3)\).
Question 2
The midpoint of \(\overline{CD}\) is \((2, -1)\) and \(C = (7, 3)\). Find \(D\).
Show the answer
Solve \(\frac{7 + x}{2} = 2\) to get \(x = -3\), and \(\frac{3 + y}{2} = -1\) to get \(y = -5\). So \(D = (-3, -5)\).
Midpoint Formula vs. Distance Formula
The midpoint formula returns a point, an ordered pair, and it adds coordinates before halving. The distance formula returns a single length, built from subtracted coordinates, squared, added and square-rooted. For \((-3, 1)\) and \((9, 7)\), the first gives \((3, 4)\) and the second gives \(\sqrt{180}\).
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