| Operation | Sample Inputs | Expected Output |
|---|---|---|
| Distance | A(2, 1), B(7, 4) | d = √((7−2)² + (4−1)²) = √34 ≈ 5.830 |
| Midpoint | A(2, 1), B(7, 4) | M = (4.5, 2.5) |
| Slope | A(2, 1), B(7, 4) | m = (4−1)/(7−2) = 0.6 |
| Triangle Area | A(0, 0), B(6, 0), C(0, 4) | Area = 12 |
| Circle (center, r) | Center (1, −3), r = 5 | (x−1)² + (y+3)² = 25 |
| Line Intersection | 2x − y + 3 = 0 and x + 2y − 4 = 0 | (x, y) = (1, 5/2) = (1, 2.5) |
Formula Used
- Distance:
d = √((x₂−x₁)² + (y₂−y₁)²) - Midpoint:
M = ((x₁+x₂)/2, (y₁+y₂)/2) - Slope:
m = (y₂−y₁)/(x₂−x₁) - Section (m:n):
P = ((n x₁ + m x₂)/(m+n), (n y₁ + m y₂)/(m+n))
- Line (point-slope):
y − y₁ = m(x − x₁) - Intersection: solve two
a x + b y + c = 0equations (Cramer’s rule) - Point-to-line distance:
|ax₀+by₀+c|/√(a²+b²) - Circle:
(x−h)² + (y−k)² = r²
- Area:
A = 1/2 |x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)| - Centroid:
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) - Circle through 3 points: center computed by determinants; fails when points are collinear.
How to Use This Calculator
- Select an operation from the dropdown.
- Enter the required coordinates or coefficients shown in the form.
- Press Calculate to view outputs and steps above the form.
- Use Download CSV or Download PDF after a successful run.
- Click Reset to clear inputs while keeping the page open.
Plane Coordinate Geometry in Practice
Plane coordinate geometry links algebra to shapes on a flat grid. This calculator covers twelve common operations, including distance, slope, midpoint, section ratio, line equations, triangle area and centroid, circle equations, line intersections, and point-to-line distance. Results are shown with step-by-step working, plus CSV and PDF exports for notes and reports. Use it for homework, drafting, and quick checks.
Distance and midpoint from two points
With A(2, 1) and B(7, 4), the changes are Δx = 7−2 = 5 and Δy = 4−1 = 3. Distance uses d = √(Δx²+Δy²) = √(25+9) = √34 ≈ 5.830. Midpoint uses M = ((x₁+x₂)/2, (y₁+y₂)/2) = (4.5, 2.5).
Slope, vertical lines, and rate of change
Slope is m = (y₂−y₁)/(x₂−x₁). For A(2, 1) to B(7, 4), m = 3/5 = 0.6, meaning y increases 0.6 units per 1 unit of x. If x₂ = x₁, the run becomes zero, so slope is undefined and the line is vertical: x = x₁.
Line equations in multiple forms
The two-point option produces point-slope, slope-intercept, and general form. For A(2, 1) and slope m = 0.6, point-slope is y − 1 = 0.6(x − 2). Converting gives y = 0.6x − 0.2, and general form 3x − 5y − 1 = 0.
Triangle area and centroid data
Triangle area is computed from coordinates using the determinant-style formula. For A(0,0), B(6,0), C(0,4), the area equals 12 square units. The centroid averages vertices: G = ((0+6+0)/3, (0+0+4)/3) = (2, 1.333…). These values help in balancing problems and region checks.
Circle equations and three-point circles
A circle with center (1, −3) and radius 5 is (x−1)² + (y+3)² = 25. The expanded form is x² + y² − 2x + 6y − 15 = 0, useful for comparisons and substitutions. The three-point circle option finds a unique circle unless the three points are collinear.
Intersections and point-to-line distance
For lines 2x − y + 3 = 0 and x + 2y − 4 = 0, the intersection is (1, 2.5). The distance from P(4, −2) to line 2x − y + 3 = 0 is |2·4 − (−2) + 3|/√(2²+(-1)²) = 13/√5 ≈ 5.817. Keep units consistent and avoid early rounding for cleaner results.
FAQs
1) What inputs are required for each operation?
Pick an operation first. The form then shows only the fields needed, such as two points, three points, a slope, or line coefficients a, b, and c.
2) Why do I get “collinear points” for the three-point circle?
If the three points lie on one straight line, infinitely many circles could fit, so no unique circle exists. The calculator detects this and stops the circle computation.
3) How do I enter a line for intersection or distance?
Use the format ax + by + c = 0. For example, 2x − y + 3 = 0 means a=2, b=-1, c=3.
4) Are results exact or rounded?
Calculations use floating-point math. Outputs display up to six decimals and trim trailing zeros. If you need more precision, enter more precise inputs and copy the steps into a higher-precision workflow.
5) What does the section ratio m:n represent?
It means AP:PB = m:n for a point P dividing segment AB internally. Larger m moves P closer to B, while larger n moves P closer to A.
6) Do units matter for coordinates?
Yes, but the tool does not assume a unit. If your coordinates are meters, distances are meters and areas are square meters. Always keep a single unit system across all inputs.