Triangle Vertices Calculator
Compute comprehensive triangle properties from three vertex coordinates including sides, angles, area (shoelace formula), perimeter, centroid, incenter, circumcenter, and orthocenter.
Triangle Slope Calculator
Calculate the slopes of all three sides of a triangle from vertex coordinates. Determine perpendicularity, parallelism, and classify the triangle by its slope properties.
Vertex A
Vertex B
Vertex C
Triangle Type⧉
Scalene Right
Classification based on side lengths and slope analysis
Right Angle?⧉
✅ Yes — AB ⊥ CA
Perpendicular sides indicate a 90° angle
Area⧉
6.0000
Computed via the shoelace formula from vertex coordinates
Perimeter⧉
12.0000
Sum of all three side lengths
Angles (approx)⧉
A: 90.00° B: 36.87° C: 53.13°
Interior angles at each vertex
Perpendicular Pairs⧉
AB ⊥ CA
Pairs of sides whose slopes multiply to −1
Slope Visualization
AB: 0.0000
|m| = 0.0000
BC: -0.7500
|m| = 0.7500
CA: Undefined (vertical)
|m| = ∞
Side Lengths
AB: 4.0000
BC: 5.0000
CA: 3.0000
Side Details
| Side | Slope | Length | Line Equation |
|---|---|---|---|
| AB | 0.0000 | 4.0000 | y = 0.0000x + 0.0000 |
| BC | -0.7500 | 5.0000 | y = -0.7500x + 3.0000 |
| CA | Undefined | 3.0000 | x = 0.00 |
Slope Reference
| Slope Value | Direction | Meaning |
|---|---|---|
| m > 0 | ↗ Rising | Line goes up from left to right |
| m < 0 | ↘ Falling | Line goes down from left to right |
| m = 0 | → Horizontal | Flat line, no vertical change |
| Undefined | ↑ Vertical | Straight up/down, no horizontal change |
| m₁ × m₂ = −1 | ⊥ | Lines are perpendicular (90°) |
| m₁ = m₂ | ∥ | Lines are parallel (never meet) |
Planning notes, formulas, and examples
About the Triangle Slope Calculator
The slope of a line measures its steepness and direction, defined as the ratio of vertical change (rise) to horizontal change (run) between any two points. When applied to a triangle defined by three vertex coordinates, the slope formula reveals critical geometric relationships among the triangle's sides.
By calculating the slopes of all three sides, you can determine important properties: whether any sides are parallel (equal slopes), whether any sides are perpendicular (slopes whose product is −1), and what type of triangle the vertices form. A right triangle, for example, will have exactly one pair of perpendicular sides. An isosceles triangle might show symmetric slope relationships.
This calculator takes three vertex coordinates in the Cartesian plane and computes the slope of each side, the corresponding line equations, side lengths, and angles. It checks for perpendicularity and parallelism among all pairs of sides, identifies the triangle type based on slope analysis, and presents the results in a clear table with visual slope bars. Whether you're working on coordinate geometry problems, verifying constructions, or analyzing triangles on a graph, the page gives you detailed slope analysis from the same three points.
When This Page Helps
Slope-based triangle analysis is useful whenever a geometry problem is written in coordinates instead of side-angle form. By computing the slope, length, and equation of each side together, this calculator makes it easy to spot right angles, vertical and horizontal edges, and collinearity issues that are easy to miss from a graph alone. It is particularly helpful in coordinate proofs where you need to justify statements about perpendicular or parallel lines with exact values.
How to Use the Inputs
- Enter the x and y coordinates for vertex A.
- Enter the x and y coordinates for vertex B.
- Enter the x and y coordinates for vertex C.
- Click a preset button to load a common triangle configuration.
- Review the slope, length, and equation of each side.
- Check the perpendicularity and parallelism results below the outputs.
Formula used
Slope m = (y₂ − y₁) / (x₂ − x₁).
Perpendicular: m₁ × m₂ = −1.
Parallel: m₁ = m₂.
Line equation: y − y₁ = m(x − x₁).
Side length = √((x₂−x₁)² + (y₂−y₁)²).Example Calculation
Result: AB has slope 0, AC is vertical, and the triangle is right-angled at A.
Vertices A(0,0), B(4,0), C(0,3). Slope AB = (0−0)/(4−0) = 0 (horizontal). Slope BC = (3−0)/(0−4) = −0.75. Slope AC = (3−0)/(0−0) = undefined (vertical). Sides AB and AC are perpendicular (horizontal × vertical). This confirms a right triangle at vertex A.
Tips & Best Practices
- A vertical side has an undefined slope — the calculator displays it as "Undefined".
- A horizontal side has a slope of exactly 0.
- Two lines are perpendicular when the product of their slopes equals −1.
- If no two sides are perpendicular, the triangle has no right angle.
- Use integer coordinates for cleaner results when practicing.
Reading Geometry from Coordinates
A triangle drawn on the coordinate plane carries more information than just its side lengths. The slopes of its edges reveal direction, steepness, and special cases such as horizontal and vertical sides. That matters in coordinate proofs, where showing that one slope is $0$ and another is undefined is enough to justify a right angle without measuring anything visually.
Perpendicular and Parallel Checks
Because the calculator reports all three line equations, it becomes easier to compare pairs of sides and verify structural relationships. Two equal slopes would indicate parallel lines, while a horizontal side paired with a vertical side confirms perpendicularity immediately. Even when slopes are ordinary real numbers, the product test for negative reciprocals helps classify the triangle more rigorously than a sketch can.
Useful for Graphing and Proofs
This kind of output is especially helpful when checking textbook coordinate-geometry exercises. You can enter three points, confirm that they are not collinear, compare approximate angles, and see whether a claimed right or isosceles triangle is actually supported by the coordinates. That saves time during proofs and gives a clean numerical check before you write a formal argument.
Sources & Methodology
Last updated:
Frequently Asked Questions
Slope measures steepness: positive slopes rise left to right, negative slopes fall left to right, zero slope means horizontal, and undefined slope means vertical.
Check each pair of sides. If the product of two slopes equals −1, those sides are perpendicular and the triangle has a 90° angle at their intersection vertex.
No. If two sides of a triangle were parallel, the three vertices would be collinear (on one line) and wouldn't form a triangle. The calculator warns you if the points are collinear.
A vertical side has an undefined slope (division by zero). The calculator handles this case and still checks perpendicularity — a vertical and horizontal line are always perpendicular.
Using point-slope form y − y₁ = m(x − x₁), which is then rearranged to slope-intercept form y = mx + b when the slope is defined.
No. The slope between two points is the same regardless of which point is first. The calculator always produces the same three sides regardless of labeling.
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