Example 1 — 2D midpoint
\(A(2,5)\), \(B(8,-1)\). \(\;M\big(\tfrac{2+8}{2},\tfrac{5+(-1)}{2}\big)=(5,2)\).
Check: \(d(M,A)=\sqrt{(5-2)^2+(2-5)^2}=\sqrt{18}=3\sqrt{2}\); \(d(M,B)=\sqrt{(8-5)^2+(-1-2)^2}=\sqrt{18}\).
Midpoint Formula Calculator
For points A(x₁, y₁) and B(x₂, y₂), the midpoint is
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ). Enter any two points and tap Calculate.
x-coordinate of the first point.
y-coordinate of the first point.
x-coordinate of the second point.
y-coordinate of the second point.
For displayed results only.
Helping Notes
- You can enter negative or decimal coordinates.
- The midpoint is the average of x’s and y’s:
((x₁+x₂)/2, (y₁+y₂)/2). - If both points are the same, the midpoint equals that point.
Result
Equation Preview
We use
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ). The preview shows your values substituted into the formula.
Midpoint
Coordinates are rounded only for display. Increase decimal places for more precision.
Extras
Extra checks include the vector from A to B and its half-step to the midpoint.
What is a Midpoint Formula Calculator?
A Midpoint Formula Calculator computes the exact halfway point between two points in a plane or in space. Given endpoints \(A(x_1,y_1)\) and \(B(x_2,y_2)\), the midpoint \(M\) lies exactly at the center of segment \(\overline{AB}\), equidistant from both ends. The same idea extends naturally to three dimensions with \(A(x_1,y_1,z_1)\), \(B(x_2,y_2,z_2)\). This tool accepts integers, decimals, and fractions, and shows each arithmetic step so learners can follow the process. It also provides quick checks—like verifying equal distances from \(M\) to both endpoints—to build intuition and confirm correctness. All formulas render responsively with MathJax or math.js for clean, readable math on any device.
About the Midpoint Formula Calculator
The midpoint is the coordinate-wise average of the endpoints. In vector form, \(M=\tfrac{A+B}{2}\). This simple average has powerful geometric meaning: it bisects the segment and is the center of symmetry between the two points. The calculator normalizes inputs (e.g., converts mixed numbers to improper fractions), computes component-wise means, and returns simplified exact results when possible. For numeric data, it also displays decimal approximations. An optional extension shows the internal-division formula (ratio form), illustrating that the midpoint is a special case of dividing a segment into a \(1{:}1\) ratio. The tool is useful in coordinate geometry, graphics programming, CAD, and physics problems that require centers, midpoints, or balanced locations between two positions.
How to Use this Midpoint Formula Calculator
- Enter endpoint \(A(x_1,y_1)\) and endpoint \(B(x_2,y_2)\). For 3D, include \(z\)-coordinates.
- Press calculate to compute the midpoint \(M\) and view step-by-step arithmetic for each coordinate.
- (Optional) Enable the equidistance check to confirm \(MA=MB\) using the distance formula.
- (Optional) Show the ratio (internal division) view to see midpoint as the special case \(1{:}1\).
- Copy the LaTeX or values into your notes, assignment, or app.
Core Formulas (LaTeX for MathJax/math.js)
2D midpoint: \[ M\!\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right) \]
3D midpoint: \[ M\!\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2},\ \frac{z_1+z_2}{2}\right) \]
Vector form: \[ \mathbf{M}=\frac{\mathbf{A}+\mathbf{B}}{2} \]
Equidistance check (2D/3D): \[ d(M,A)=d(M,B)=\sqrt{\sum (\,\text{coord}_M-\text{coord}_{A/B}\,)^2} \]
Internal division (ratio \(r:s\)): \[ P=\left(\frac{r x_2+s x_1}{r+s},\ \frac{r y_2+s y_1}{r+s}\right)\quad\text{(midpoint when }r=s=1\text{)}. \]
Examples (Illustrative)
Example 2 — 3D midpoint
\(A(1,4,2)\), \(B(5,-2,6)\). \(\;M\big(\tfrac{1+5}{2},\tfrac{4+(-2)}{2},\tfrac{2+6}{2}\big)=(3,1,4)\).
Example 3 — Fractional inputs
\(A\!\left(\tfrac{1}{2},\tfrac{3}{4}\right)\), \(B\!\left(\tfrac{5}{2},\tfrac{7}{4}\right)\). \(\;M\!\left(\tfrac{1/2+5/2}{2},\tfrac{3/4+7/4}{2}\right)=\left(\tfrac{3}{2},\tfrac{5}{4}\right)\).
FAQs
What is the midpoint formula?
The coordinate-wise average of endpoints: \(M\big(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\big)\) in 2D; extend with \(z\) for 3D.
Does the calculator support 3D points?
Yes. Enter \(z\)-coordinates to compute \(M\!\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2},\tfrac{z_1+z_2}{2}\right)\).
Can I use fractions or negative numbers?
Absolutely. The tool simplifies exact fractions and also shows decimal approximations.
Is the midpoint always equidistant from endpoints?
Yes. By definition, \(d(M,A)=d(M,B)\); the calculator can display this check.
How do I find a midpoint on a number line?
Use \(m=\tfrac{a+b}{2}\) since 1D is just a special case of the 2D formula.
What if both endpoints are the same?
The midpoint equals that same point; the segment length is zero.
Does the order of points matter?
No. Averaging is symmetric: swapping endpoints yields the same midpoint.
Can this help find a segment’s center for a circle chord?
Yes. The midpoint of a chord is its center along the chord and lies on the circle’s perpendicular bisector.
How precise are results?
Exact arithmetic is used for rationals; decimals are rounded to your chosen precision if enabled.
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