Midpoint Calculator

Enter two points in the coordinate plane and instantly get the midpoint, individual coordinates, and the formula used.

Coordinate Points

Point 1 (x, y)

Point 2 (x, y)

Midpoint Results

Midpoint M (0, 0)

X-coordinate 0

Y-coordinate 0

Formula Used M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )

Midpoint Formula: Complete Guide with Geometry, Examples, and Applications

The midpoint of a segment is the point exactly halfway between two endpoints in the coordinate plane, useful in geometry, analytic proofs, and practical design tasks.

This guide follows the same pattern as your exponent page: explanation boxes, step-by-step examples, real-world uses, pitfalls, and practice problems, all centered on midpoint concepts.

What Is the Midpoint?

Given two points A(x₁, y₁) and B(x₂, y₂), the midpoint M of segment AB is the point that lies exactly in the middle of A and B on both x and y axes.

Key Components

Endpoints: A(x₁, y₁) and B(x₂, y₂) that define the segment.
Midpoint: M( (x₁+x₂)/2 , (y₁+y₂)/2 ), the average of x‑coordinates and y‑coordinates.
Segment: The straight line connecting A and B in the coordinate plane.

Example 1: Simple midpoint
A(2, 4) and B(6, 8).
M = ( (2+6)/2 , (4+8)/2 ) = (8/2, 12/2) = (4, 6).

Example 2: Negative coordinates
A(−3, 5) and B(5, −1).
M = ( (−3+5)/2 , (5+(−1))/2 ) = (2/2, 4/2) = (1, 2).

Point A	Point B	Midpoint M	Notes
(0, 0)	(2, 2)	(1, 1)	Diagonal from origin
(−4, 2)	(4, 2)	(0, 2)	Horizontal segment
(1, −3)	(5, 1)	(3, −1)	Mixed signs
(−2, −2)	(2, 2)	(0, 0)	Symmetric about origin
(3, 7)	(3, −1)	(3, 3)	Vertical segment

Deriving and Using the Midpoint Formula

The midpoint formula comes from averaging coordinates: the x‑coordinate of the midpoint is halfway between x₁ and x₂, and the y‑coordinate is halfway between y₁ and y₂.

Midpoint formula in 2D

M( (x₁+x₂)/2 , (y₁+y₂)/2 )

Example:
A(−1, 4) and B(7, 0).
M = ( (−1+7)/2 , (4+0)/2 ) = (6/2, 4/2) = (3, 2).

Connection with averages

The midpoint coordinates are simply the arithmetic means of the corresponding coordinates of A and B.

Example:
Average of x‑values (2 and 10) is (2+10)/2 = 6; average of y‑values (−4 and 8) is (−4+8)/2 = 2. Midpoint is (6, 2).

Extension to 3D

For points A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the midpoint is M( (x₁+x₂)/2 , (y₁+y₂)/2 , (z₁+z₂)/2 ).

Example:
A(1, 2, 3) and B(5, −2, 7).
M = ( (1+5)/2 , (2+(−2))/2 , (3+7)/2 ) = (3, 0, 5).

Dimension	Endpoints	Formula	Midpoint
1D	x₁, x₂	(x₁+x₂)/2	Point on number line
2D	(x₁, y₁), (x₂, y₂)	((x₁+x₂)/2, (y₁+y₂)/2)	Point in plane
3D	(x₁, y₁, z₁), (x₂, y₂, z₂)	((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)	Point in space

Step-by-Step Worked Examples

These examples walk through midpoint calculations in detail; you can confirm each result instantly using the Midpoint Calculator above.

Example 1: Segment in first quadrant

Find the midpoint of A(1.5, 3.5) and B(4.5, 7.5).

Solution:
Step 1: Compute x‑coordinate: (1.5 + 4.5)/2 = 6/2 = 3.

Step 2: Compute y‑coordinate: (3.5 + 7.5)/2 = 11/2 = 5.5.

Step 3: Midpoint is M(3, 5.5).

Example 2: Mixed signs

Find the midpoint of A(−2, 5) and B(6, −3).

Solution:
Step 1: x‑coordinate: (−2 + 6)/2 = 4/2 = 2.

Step 2: y‑coordinate: (5 + (−3))/2 = 2/2 = 1.

Step 3: M(2, 1).

Example 3: Using midpoint to find an endpoint

A has coordinates (4, −1) and the midpoint M is (1, 2). Find B.

Solution:
Step 1: Use formula: (x₁+x₂)/2 = midpoint x, (y₁+y₂)/2 = midpoint y.

Step 2: For x: (4 + x₂)/2 = 1 ⇒ 4 + x₂ = 2 ⇒ x₂ = −2.

Step 3: For y: (−1 + y₂)/2 = 2 ⇒ −1 + y₂ = 4 ⇒ y₂ = 5.

Step 4: B is (−2, 5).

Real-World Applications of the Midpoint

Midpoints appear whenever something must be centered, balanced, or positioned halfway between two locations, both in math problems and real projects.

📐 Geometry and Constructions

Finding midpoints is essential for constructing perpendicular bisectors, medians of triangles, and symmetry lines in coordinate geometry.

🏗️ Engineering and Design

Engineers use midpoints to place supports, determine center points for beams, and align structural elements between two given coordinates.

🗺️ Mapping and Navigation

Midpoints help find central meeting locations between two coordinates, or approximate mid-latitude and longitude positions in simple models.

🎮 Game Development

Games use midpoint logic to position objects halfway between players, calculate camera focus points, and center UI elements dynamically.

📊 Data Visualization

Midpoints between data points support interpolation, smoothing, and constructing midline curves in charts and graphs.

🎨 Graphic Design

Design tools frequently use midpoints to align shapes, center text between guides, and snap objects to the halfway point of a selection.

Common Mistakes and How to Avoid Them

Most midpoint mistakes come from mixing up formulas, misplacing signs, or averaging incorrectly. Careful setup removes these errors.

❌ MISTAKE 1: Using distance instead of average
Wrong: M = (x₂−x₁)/2, (y₂−y₁)/2.
✅ Correct: M = (x₁+x₂)/2, (y₁+y₂)/2.

❌ MISTAKE 2: Forgetting negative signs
Dropping minus signs when adding coordinates shifts the midpoint to the wrong quadrant.
✅ Carefully add signed numbers before dividing by 2.

❌ MISTAKE 3: Dividing only one coordinate
Averaging x but not y gives a point that is not centered on the segment.
✅ Always average both x and y coordinates.

❌ MISTAKE 4: Rounding too early
Rounding intermediate steps can slightly move the midpoint.
✅ Keep extra decimal places and round only in the final answer or output display.

Midpoint, Distance, and Slope

The midpoint formula works together with distance and slope formulas to describe segments completely in coordinate geometry.

Key formulas for a segment AB

Midpoint: M( (x₁+x₂)/2 , (y₁+y₂)/2 )
Distance: d = √( (x₂−x₁)² + (y₂−y₁)² )
Slope: m = (y₂−y₁)/(x₂−x₁), if x₂≠x₁

Example:
A(1, 2), B(5, 6).
Midpoint: ( (1+5)/2 , (2+6)/2 ) = (3, 4).
Distance: √(4² + 4²) = √32 = 4√2.
Slope: (6−2)/(5−1) = 4/4 = 1.

Quantity	Formula	What it tells you
Midpoint	((x₁+x₂)/2, (y₁+y₂)/2)	Center of the segment
Distance	√((x₂−x₁)² + (y₂−y₁)²)	Length of the segment
Slope	(y₂−y₁)/(x₂−x₁)	Steepness and direction of the line

Practice Problems with Solutions

Try these midpoint questions, then reveal the solutions and verify the coordinates using the Midpoint Calculator above.

Basic Level

Problem 1: Find the midpoint of A(2, 6) and B(8, 10).

Show Solution

M = ( (2+8)/2 , (6+10)/2 ) = (10/2, 16/2) = (5, 8)

Problem 2: Find the midpoint of A(−4, −2) and B(4, 6).

Show Solution

M = ( (−4+4)/2 , (−2+6)/2 ) = (0/2, 4/2) = (0, 2)

Intermediate Level

Problem 3: A(5, −1) and B(−3, 7). Find M.

Show Solution

M = ( (5+(−3))/2 , (−1+7)/2 ) = (2/2, 6/2) = (1, 3)

Problem 4: The midpoint between A(−1, 4) and B(x, y) is (2, 1). Find B.

Show Solution

(−1 + x)/2 = 2 ⇒ −1 + x = 4 ⇒ x = 5.
(4 + y)/2 = 1 ⇒ 4 + y = 2 ⇒ y = −2.
So B is (5, −2).

Advanced Level

Problem 5: Points A and B have midpoint (0, 0), and A is (7, −5). Find B.

Show Solution

(7 + x)/2 = 0 ⇒ 7 + x = 0 ⇒ x = −7.
(−5 + y)/2 = 0 ⇒ −5 + y = 0 ⇒ y = 5.
B is (−7, 5).

Problem 6: A(1, 2, 3) and B(5, 6, 7) in 3D. Find M.

Show Solution

M = ( (1+5)/2 , (2+6)/2 , (3+7)/2 ) = (3, 4, 5)