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Use triangle angle-side relationships and triangle inequality in mathematical and real-world problems.

Classify three lengths with the strict triangle inequality

Problem
Classify whether positive side lengths 3, 4, and 5 form a nondegenerate triangle.
Three candidate segments with positive lengths 3, 4, and 5; no triangle or inequality conclusion is shown. Open full size
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Hint

Sort the three lengths and add the two shorter ones.

For positive sorted sides a≤b≤c, a nondegenerate triangle exists exactly when a+b>c.

Solution walkthrough

01

Sort and identify the decisive side

\[3\le~4\le~5;~\text{longest}~\text{side}=5\]

For positive side lengths, sorting allows the triangle inequality to be checked with the two shorter sides against the longest. The other two inequalities then hold automatically.

02

Apply the strict closure condition

\[3+4>5\]

A nondegenerate triangle exists exactly when the two shorter lengths add to more than the longest length.

03

Evaluate the comparison

\[3+4=7~\text{and}~7>5\]

The two shorter segments total 7, which exceeds 5 by 2, so they can meet rather than lying flat or leaving a gap.

04

Classify and distinguish the boundary

\[\text{strict}~\text{inequality}~->~\text{yes},~\text{nondegenerate}~\text{triangle}\]

Equality would make a straight degenerate figure, and a smaller sum would fail to close. Because the inequality is strict, these lengths form a genuine triangle.

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Another way

  1. Verify all three sums: 3+4>5, 3+5>4, and 4+5>3.

!

Common mistake

Do not merely note that all lengths are positive. Positive lengths still must satisfy the strict triangle inequality.

A scalene triangle with side lengths 3, 4, and 5. It shows that 3 plus 4 is greater than 5, so the segments can form a triangle.
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