Calculator Inputs
Enter one point and choose the line format that defines the mirror.
Formula Used
Mirror line in standard form: ax + by + c = 0
Reflection formulas: x′ = x - 2a(ax + by + c)/(a² + b²), y′ = y - 2b(ax + by + c)/(a² + b²)
Distance from the point to the line: |ax + by + c| / √(a² + b²)
The calculator first converts your chosen line format into standard form. It then projects the point onto the line and moves the same distance beyond the line along the normal direction to find the reflected image.
How to Use This Calculator
- Enter the original point coordinates you want to reflect.
- Choose the line input mode that best matches your question.
- Fill in the line values, such as coefficients, intercepts, constants, or two points.
- Select the decimal precision for cleaner classroom or exam outputs.
- Press Submit to show the reflected point above the form.
- Use the export buttons to save the result table as CSV or PDF.
Example Data Table
These sample cases show how different line formats affect the reflected image.
| Original Point | Mirror Line | Reflected Point | Note |
|---|---|---|---|
| (4, 2) | y = x | (2, 4) | Coordinates swap across the diagonal. |
| (5, -1) | x = 2 | (-1, -1) | Horizontal position flips around x = 2. |
| (3, 6) | y = 1 | (3, -4) | Vertical distance stays equal on both sides. |
| (1, 4) | x + y - 3 = 0 | (-1, 2) | Standard-form lines work without conversion by hand. |
FAQs
1. What does reflecting a point over a line mean?
It creates a new point on the opposite side of the line at exactly the same perpendicular distance. The mirror line becomes the midpoint path between the original point and its reflected image.
2. Which line formats does this calculator accept?
It accepts standard form, slope-intercept form, vertical lines, horizontal lines, and lines defined by two points. Each format is converted internally into the same standard reflection model.
3. Why is standard form useful for reflection problems?
Standard form makes the reflection formula direct because the line normal is already visible in the coefficients a and b. That helps compute distance, projection, and reflected coordinates cleanly.
4. What happens if the point lies on the mirror line?
The reflected point is the same as the original point. In that case, the shortest distance to the line is zero, and the midpoint and foot of the perpendicular match the same coordinates.
5. Does the calculator show more than the reflected point?
Yes. It also reports the midpoint, foot of the perpendicular, distance to the line, segment length, slope information, and a short explanation of the symmetry result.
6. Why might my line entry show an error?
Errors appear when required values are missing, when the line definition is invalid, or when two points used for the line are identical. The line must describe one real mirror line.
7. Can I use decimals or negative values?
Yes. The inputs accept integers, decimals, fractions entered as decimals, and negative values. You can also control the displayed rounding by choosing the decimal precision before submitting.
8. What are the CSV and PDF downloads for?
They let you keep a clean copy of the current result for notes, assignments, tutoring, or review. CSV works well for spreadsheets, while PDF is better for sharing or printing.