In a triangle, the lengths of two sides are 8 and 13. Which of the following could be the length...

1. INFER what mathematical principle applies

• This is a triangle with two known sides (8 and 13) and one unknown side
• The triangle inequality theorem must be satisfied for a valid triangle to exist
• Key insight: All three combinations of "sum of two sides > third side" must be true

2. INFER the strategy and set up inequalities

• Let \(\mathrm{x}\) = length of unknown third side
• Apply triangle inequality in all three ways:
  • \(\mathrm{8 + 13 \gt x}\) (sum of known sides > unknown side)
  • \(\mathrm{8 + x \gt 13}\) (first known side + unknown > second known side)
  • \(\mathrm{13 + x \gt 8}\) (second known side + unknown > first known side)

3. SIMPLIFY each inequality

• From \(\mathrm{8 + 13 \gt x}\): \(\mathrm{21 \gt x}\), so \(\mathrm{x \lt 21}\)
• From \(\mathrm{8 + x \gt 13}\): \(\mathrm{x \gt 13 - 8}\), so \(\mathrm{x \gt 5}\)
• From \(\mathrm{13 + x \gt 8}\): \(\mathrm{x \gt 8 - 13}\), so \(\mathrm{x \gt -5}\)
• Since side lengths are positive, the third constraint is automatically satisfied

4. APPLY CONSTRAINTS to find the valid range

• Combining the meaningful constraints: \(\mathrm{x \gt 5}\) AND \(\mathrm{x \lt 21}\)
• This gives us: \(\mathrm{5 \lt x \lt 21}\)
• Important: The inequality is strict (< and >), not inclusive (≤ and ≥)

5. APPLY CONSTRAINTS to evaluate answer choices

• (A) 5: Invalid because 5 is NOT greater than 5
• (B) 19: Valid because \(\mathrm{5 \lt 19 \lt 21}\) ✓
• (C) 21: Invalid because 21 is NOT less than 21
• (D) 22: Invalid because 22 is NOT less than 21

Answer: B (19)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often remember the triangle inequality theorem but only apply one version: "the sum of two sides must be greater than the third." They set up just \(\mathrm{8 + 13 \gt x}\), getting \(\mathrm{x \lt 21}\), then select any answer choice less than 21.

This leads them to incorrectly think that both 5 and 19 are valid, creating confusion about which to choose. They might select Choice A (5) because it seems "safer" being farther from the boundary.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students correctly derive \(\mathrm{5 \lt x \lt 21}\) but incorrectly treat the inequality as inclusive (≤), thinking that exactly 5 or exactly 21 could work as side lengths.

This may lead them to select Choice C (21) since they incorrectly believe the boundary value is acceptable.

The Bottom Line:

Triangle inequality problems require systematic application of ALL three inequality constraints, not just the most obvious one. The key insight is recognizing that strict inequalities exclude the boundary values completely.