Calculator Input
Point Symmetry Graph
The graph plots original points, symmetric points, and the center of symmetry.
Formula Used
Point symmetry across a center C(h, k) maps each point P(x, y) to a new point P′(x′, y′). The center becomes the midpoint of the original and reflected point.
Main formula: x′ = 2h − x and y′ = 2k − y
Midpoint check: ((x + x′) / 2, (y + y′) / 2) = (h, k)
Distance formula: d = √[(x′ − x)² + (y′ − y)²]
For segments and polygons, apply the same coordinate rule to every endpoint or vertex.
How to Use This Calculator
- Select whether you want a single point, line segment, or polygon reflection.
- Enter the center of symmetry coordinates h and k.
- Input the original coordinates for the chosen object.
- Choose the number of decimal places for displayed values.
- Press Calculate Symmetry to generate the reflected coordinates.
- Review the result summary above the form and the graph below.
- Use the CSV or PDF buttons to export your results.
Example Data Table
| Case | Center | Original Object | Symmetric Result |
|---|---|---|---|
| Single Point | C(0, 0) | P(3, 5) | P′(-3, -5) |
| Single Point | C(2, 1) | P(5, 7) | P′(-1, -5) |
| Segment | C(1, 1) | A(2, 3), B(5, 4) | A′(0, -1), B′(-3, -2) |
| Polygon | C(2, 2) | (1,1), (4,2), (3,5) | (3,3), (0,2), (1,-1) |
FAQs
1. What is point symmetry?
Point symmetry means every point on a figure has a matching point directly opposite a center. The center is the midpoint of each original and reflected pair.
2. How is point symmetry different from line reflection?
Point symmetry reflects coordinates through a center point. Line reflection mirrors coordinates across a line, such as the x-axis, y-axis, or y = x.
3. Can this calculator reflect polygons?
Yes. Enter polygon vertices one per line as x,y. The calculator reflects every vertex around the chosen center and plots both shapes.
4. Why does the center become the midpoint?
In point symmetry, the original point and its image lie equally far from the center in opposite directions. That makes the center their exact midpoint.
5. Can decimal values be used?
Yes. The calculator accepts integers and decimals for center coordinates, points, and polygon vertices. You can also control rounding precision.
6. Does a symmetric segment keep the same length?
Yes. Point symmetry is a rigid transformation. It preserves segment length, angle measures, and overall figure shape while changing position.
7. What is the formula for a reflected point?
If the center is C(h, k) and the point is P(x, y), then the reflected point is P′(2h − x, 2k − y).
8. What export options are included?
The tool includes CSV export for numeric results and a PDF print export using the browser print dialog for quick saving and sharing.