1. What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that for any three lengths to form a triangle, the sum of any two sides must be greater than the third side. This fundamental principle governs all possible triangles in Euclidean geometry.

2. How the Theorem Works

The theorem requires all three conditions to be true simultaneously:

Where:

• \( a, b, c \) — Lengths of the three sides

Explanation: If any one of these inequalities fails, the three lengths cannot form a triangle. The theorem ensures that the sides can "close" to form a proper triangle.

3. Practical Applications

Details: This theorem is essential in geometry, architecture, engineering, and computer graphics. It helps determine if three points can form a triangle, or if three lengths can construct a valid triangular structure.

4. Using the Calculator

Tips: Enter three positive lengths in any units. The calculator will check all three conditions of the theorem and determine if a triangle is possible.

5. Frequently Asked Questions (FAQ)

Q1: Can the sides be equal?
A: Yes, equal sides form special triangles (equilateral or isosceles) as long as they satisfy the inequality conditions.

Q2: What about degenerate triangles?
A: If the sum equals (rather than exceeds) the third side, the points are colinear (a degenerate triangle).

Q3: Does this work for all triangle types?
A: Yes, the theorem applies to all triangles - acute, right, and obtuse.

Q4: How precise must the measurements be?
A: The theorem is mathematically exact. In practical applications, measurement precision affects real-world applicability.

Q5: Can this determine triangle type?
A: No, this only determines possibility. Additional calculations would determine if the triangle is acute, right, or obtuse.