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Midpoint Formula

Midpoint Formula

Midpoint formula coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2) is the formula for the midpoint between points (x1, y1) and (x2, y2). Note that this is simply the average of the x-coordinates and the average of the y-coordinates.

Midpoint formulas: on a plane: ((x1+x2)/2, (y1+y2)/2); in three dimensions: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)

 

 

See also

Point of division formula

Key Formula

M=(x1+x22,  y1+y22)M = \left( \frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2} \right)
Where:
  • MM = The midpoint of the line segment connecting the two points
  • x1x_1 = The x-coordinate of the first point
  • y1y_1 = The y-coordinate of the first point
  • x2x_2 = The x-coordinate of the second point
  • y2y_2 = The y-coordinate of the second point

Worked Example

Problem: Find the midpoint of the line segment connecting the points (2, 8) and (10, 4).
Step 1: Identify the coordinates of each point.
(x1,y1)=(2,8)and(x2,y2)=(10,4)(x_1, y_1) = (2, 8) \quad \text{and} \quad (x_2, y_2) = (10, 4)
Step 2: Average the x-coordinates by adding them and dividing by 2.
x1+x22=2+102=122=6\frac{x_1 + x_2}{2} = \frac{2 + 10}{2} = \frac{12}{2} = 6
Step 3: Average the y-coordinates by adding them and dividing by 2.
y1+y22=8+42=122=6\frac{y_1 + y_2}{2} = \frac{8 + 4}{2} = \frac{12}{2} = 6
Step 4: Combine these results to write the midpoint as an ordered pair.
M=(6,6)M = (6, 6)
Answer: The midpoint is (6, 6).

Another Example

This example works the midpoint formula in reverse — given the midpoint and one endpoint, you solve for the missing endpoint. This is a very common test question.

Problem: The midpoint of a segment is (5, 3) and one endpoint is (1, 7). Find the other endpoint.
Step 1: Write the midpoint formula and substitute the known values. Let the unknown endpoint be (x₂, y₂).
(1+x22,  7+y22)=(5,3)\left( \frac{1 + x_2}{2},\; \frac{7 + y_2}{2} \right) = (5, 3)
Step 2: Set the x-components equal and solve for x₂.
1+x22=5  ⟹  1+x2=10  ⟹  x2=9\frac{1 + x_2}{2} = 5 \implies 1 + x_2 = 10 \implies x_2 = 9
Step 3: Set the y-components equal and solve for y₂.
7+y22=3  ⟹  7+y2=6  ⟹  y2=−1\frac{7 + y_2}{2} = 3 \implies 7 + y_2 = 6 \implies y_2 = -1
Step 4: Write the other endpoint as an ordered pair.
(x2,y2)=(9,−1)(x_2, y_2) = (9, -1)
Answer: The other endpoint is (9, −1).

Frequently Asked Questions

What is the difference between the midpoint formula and the distance formula?
The midpoint formula gives you the coordinates of the point exactly halfway between two points, while the distance formula gives you the length of the segment connecting them. The midpoint formula uses addition and division (averaging), whereas the distance formula uses subtraction and the Pythagorean theorem.
Does it matter which point is (x₁, y₁) and which is (x₂, y₂)?
No. Because addition is commutative, swapping the two points gives the same averages and therefore the same midpoint. You can label either point as the first or second.
Can the midpoint formula be used in three dimensions?
Yes. For points (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z2)(x_2, y_2, z_2), the midpoint is (x1+x22,  y1+y22,  z1+z22)\left(\frac{x_1+x_2}{2},\; \frac{y_1+y_2}{2},\; \frac{z_1+z_2}{2}\right). You simply average each coordinate separately.

Midpoint Formula vs. Distance Formula

Midpoint FormulaDistance Formula
What it findsThe coordinates of the center point between two pointsThe length of the segment between two points
Formula(x1+x22,  y1+y22)\left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right)(x2−x1)2+(y2−y1)2\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Output typeAn ordered pair (a point)A single number (a distance)
Core operationAveraging (addition then division)Pythagorean theorem (subtraction, squaring, square root)
When to useFinding the center of a segment, checking if a point bisects a segmentFinding how far apart two points are, checking if segments are congruent

Why It Matters

The midpoint formula appears throughout coordinate geometry — you use it to find the center of a circle given a diameter, to locate the intersection of diagonals in a rectangle, and to verify that a point bisects a segment. It is also essential in proofs where you need to show that diagonals of a parallelogram bisect each other. Beyond geometry class, the concept of averaging coordinates extends to data analysis and computer graphics.

Common Mistakes

Mistake: Subtracting the coordinates instead of adding them.
Correction: The midpoint formula requires you to add the coordinates before dividing by 2. Subtraction is used in the distance formula, not the midpoint formula. Remember: midpoint = average, and an average uses addition.
Mistake: Forgetting to divide by 2 after adding.
Correction: Adding the coordinates without dividing by 2 gives you a point that is not the midpoint. Always complete both parts of the average: add, then divide by 2.

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