Questions: Triangle Inequality Theorem

Question 1

Side lengths of 5, 5, and 10 are given. Can these form a triangle?

• A. Yes — two sides equal the third, which satisfies the Triangle Inequality
• B. No — the two shorter sides must be strictly greater than the longest, but 5 + 5 = 10, not more than 10
• C. Yes — any three positive lengths can form a triangle
• D. No — only right triangles can have two equal sides

Answer: (B)

The Triangle Inequality requires a strict greater-than (>), not greater-than-or-equal. When 5 + 5 = 10, the three points are collinear — A, B, and C lie on a straight line and the shape collapses to a segment with zero area. This degenerate case is excluded by the strict inequality. Using ≥ instead of > is the most common error students make on this topic.

Question 2

When checking whether three lengths can form a triangle, which inequality is the only one that can actually fail?

• A. The two longer sides must exceed the shortest
• B. The two shorter sides must exceed the longest
• C. All three pairwise inequalities must be checked independently
• D. The longest side must be more than twice the shortest

Answer: (B)

If the two shorter sides sum to more than the longest, the other two inequalities follow automatically — adding the longer of the two short sides to the longest side will always exceed the remaining short side. Only the binding constraint (shortest pair vs. longest side) can fail. This is why in practice you only need to check: do the two smaller lengths sum to more than the largest?

Question 3

If a + b = c for side lengths a, b, and c (where c is the longest), then a, b, and c form a valid (non-degenerate) triangle.

• T. True
• F. False

Answer: False

When a + b = c exactly, the three points A, B, and C are collinear — the 'triangle' degenerates to a line segment with zero area. The Triangle Inequality requires strict greater-than (a + b > c), not greater-than-or-equal. This degenerate case is the most common source of error when students use ≥ instead of >.

Question 4

The Triangle Inequality implies that traveling from point A to point C by detouring through any intermediate point B will always cover more distance than going directly from A to C.

• T. True
• F. False

Answer: True

This is the geometric heart of the theorem. Going directly covers |AC|. Any detour through B covers |AB| + |BC|, which by the Triangle Inequality must be strictly greater than |AC|. The straight-line path is always the shortest. This extends beyond geometry — the Triangle Inequality is one of the axioms any well-defined distance function must satisfy in mathematics.

Question 5

Why does the Triangle Inequality use strict greater-than (>) rather than greater-than-or-equal (≥), and what happens geometrically when equality holds?

Model answer: When equality holds (a + b = c), the three vertices are collinear — they lie on a straight line, producing a degenerate 'triangle' with zero area that is geometrically just a segment. A true triangle requires the two shorter sides to actually 'reach past' the longest, not merely touch it. The strict inequality (>) excludes this degenerate case.

The distinction matters both for geometric correctness (a degenerate triangle is not a triangle) and for applying the theorem: the strict inequality is what ensures the three points form a closed polygon with positive area. Any problem that allows equality is implicitly permitting straight-line configurations, which violates the definition of a triangle.