Midpoint Formula

Learn how to find the exact center point between two coordinates on a graph.

Intermediate20 minLesson
Printable worksheet

Definition

The midpoint of a line segment is the point exactly halfway between its two endpoints.
To find the midpoint of a segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), use the midpoint formula:
M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
In other words, the midpoint is found by averaging the x-coordinates and averaging the y-coordinates of the endpoints.

Try it now

What is the midpoint of a segment with endpoints (0,0)(0, 0) and (4,6)(4, 6)?

Worked Examples

Find the midpoint of the segment with endpoints A(2,4)A(2, 4) and B(8,10)B(8, 10).

1

Identify the coordinates

x1=2x_1 = 2, y1=4y_1 = 4, x2=8x_2 = 8, y2=10y_2 = 10 → Coordinates identified

2

Average the x-coordinates

x1+x22=2+82=102=5\frac{x_1 + x_2}{2} = \frac{2 + 8}{2} = \frac{10}{2} = 5 → xM=5x_M = 5

3

Average the y-coordinates

y1+y22=4+102=142=7\frac{y_1 + y_2}{2} = \frac{4 + 10}{2} = \frac{14}{2} = 7 → yM=7y_M = 7

4

Write the midpoint

M=(5,7)M = (5, 7) → Midpoint found

Common Mistakes

Subtracting coordinates instead of adding them

Why it's wrong: The midpoint formula requires averaging (adding then dividing by 2), not finding the difference.

Correct: Always ADD the coordinates: x1+x22\frac{x_1 + x_2}{2}, not x2−x12\frac{x_2 - x_1}{2}.

Forgetting to divide by 2

Why it's wrong: Adding the coordinates gives the sum, not the average. You must divide by 2 to find the point in the middle.

Correct: Remember: midpoint means AVERAGE, so always divide the sum by 2.

Mixing up x and y coordinates

Why it's wrong: Accidentally pairing x1x_1 with y2y_2 or making calculation errors.

Correct: Work systematically: first average ALL x-values, then average ALL y-values separately.

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Practice Problems

15 problems
Problem 1 of 15
Easy

What is the midpoint of a segment with endpoints (0,0)(0, 0) and (4,6)(4, 6)?

Why It Matters

The midpoint formula is used everywhere in geometry and real life:
  • Navigation: Finding the halfway point of a journey
  • Architecture: Locating the center of walls, windows, and rooms
  • Sports: Determining the center of a playing field
  • Computer Graphics: Calculating positions for animations and game objects
It's also the foundation for more advanced concepts like finding the center of a circle or the centroid of a triangle!

Real World Applications

Meeting in the Middle

Two friends live in different towns and want to meet at a point equidistant from both.

Example:

If Alex lives at coordinates (10,20)(10, 20) and Jordan lives at (30,40)(30, 40), the meeting point is at (20,30)(20, 30).

1Try It Yourself

Emma lives at (4,2)(4, 2) and Noah lives at (12,8)(12, 8). They want to meet at a cafe exactly halfway between them.

What are the coordinates of the cafe?

Step 1: Write the mathematical expression

Use the midpoint formula:

Center of a Playing Field

Sports fields need a center point marked for kickoffs, jump balls, and face-offs.

Example:

A rectangular field has corners at (0,0)(0, 0) and (100,60)(100, 60). The center is at (50,30)(50, 30).

2Try It Yourself

A basketball court has one corner at coordinates (0,0)(0, 0) and the opposite corner at (28,15)(28, 15).

Where is the center circle located?

Step 1: Write the mathematical expression

Find the midpoint of the diagonal:

Key Takeaways

  • 1The midpoint is the point exactly halfway between two endpoints
  • 2Use the midpoint formula: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  • 3Find the midpoint by averaging the x-coordinates and averaging the y-coordinates
  • 4The formula works with positive, negative, and decimal coordinates

Frequently Asked Questions

No! Addition is commutative, so x1+x22=x2+x12\frac{x_1 + x_2}{2} = \frac{x_2 + x_1}{2}. You'll get the same midpoint either way.
No! Addition is commutative, so x1+x22=x2+x12\frac{x_1 + x_2}{2} = \frac{x_2 + x_1}{2}. You'll get the same midpoint either way.
That's perfectly fine! Midpoints often have decimal or fractional coordinates. For example, the midpoint of (0,0)(0, 0) and (3,5)(3, 5) is (1.5,2.5)(1.5, 2.5).
The midpoint formula IS an average! The x-coordinate of the midpoint is the average of the x-coordinates, and the y-coordinate is the average of the y-coordinates.

Glossary

Midpoint
The point that divides a line segment into two equal parts
Line segment
A part of a line with two endpoints
Endpoint
A point at the end of a line segment
Coordinates
An ordered pair (x,y)(x, y) that shows the position of a point on a graph
Average
The sum of values divided by the number of values

Formula Card

Midpoint Formula

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Average the x-coordinates and y-coordinates of the two endpoints to find the midpoint.

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