Malone Global University

Lesson 5.4: Triangle Inequality Theorem

Learn the rule that determines whether three side lengths can form a triangle and apply it to solve geometry problems.

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Lesson Description

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the third side. This rule allows us to determine whether a triangle is possible before solving it.

Learning Objectives

  • State the Triangle Inequality Theorem.
  • Test whether three lengths can form a triangle.
  • Find the possible range of a missing side length.
  • Apply triangle inequality to real geometric problems.

Topics Covered

  • Triangle Inequality rule
  • Checking all three inequalities
  • Determining possible side ranges
  • Geometric interpretation

Key Terms

  • Triangle Inequality Theorem: For sides \(a,b,c\): \(a+b>c,\; a+c>b,\; b+c>a\)
  • Valid Triangle: A triangle whose side lengths satisfy the inequality.
  • Degenerate Triangle: When the sum equals the third side (forms a straight line).

Lesson Examples

Example 1: Check if 3, 4, 5 form a triangle

  1. 3+4>5 → 7>5 ✓
  2. 3+5>4 → 8>4 ✓
  3. 4+5>3 → 9>3 ✓
  4. All true → triangle exists.

Example 2: Check if 2, 3, 6 form a triangle

  1. 2+3>6 → 5>6 ✗
  2. Fails → not a triangle.

Example 3: Find possible third side

  1. Sides: 8 and 11
  2. Third side \(x\)
  3. \(|11-8| < x < 11+8\)
  4. \(3 < x < 19\)

Practice Problems

  1. Determine whether 5, 7, 12 can form a triangle.
  2. Determine whether 6, 8, 10 can form a triangle.
  3. Two sides are 9 and 14. Find the integer values possible for the third side.
  4. Explain why the shortest path between two points is a straight line using triangle inequality.

Tips & Common Mistakes

  • You must check all three inequalities.
  • If the sum equals the third side, it is NOT a triangle.
  • The third side must be strictly between the difference and sum.
  • Use absolute value when finding the lower bound.

Summary

The Triangle Inequality Theorem guarantees that any triangle must have side lengths whose pairwise sums exceed the remaining side. This rule is essential for validating triangles and determining unknown side ranges.

Challenge Problems

  1. Prove the Triangle Inequality using geometric construction.
  2. Two sides measure 15 and 20. Find all integer third sides that form an obtuse triangle.
  3. Explain why walking directly between two locations is shorter than taking a two-segment route.
  4. A triangle has integer sides and perimeter 20. List all possibilities.

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