Triangle Inequality Theorem

Example Problem Setup

• Triangle with sides 6, 10, and x.
• Goal: Determine the possible range for x.

Determining the Minimum Value of x

1. Concept: Minimize x by making the angle between sides 6 and 10 as small as possible.
2. Degenerate Triangle: When the angle approaches 0, sides 6 and 10 coincide, and x = 10 - 6 = 4.
3. Conclusion: For a non-degenerate triangle, x > 4 (x must be greater than 4).

Determining the Maximum Value of x

1. Concept: Maximize x by making the angle between sides 6 and 10 as large as possible.
2. Degenerate Triangle: When the angle approaches 180 degrees, sides 6 and 10 form a straight line, and x = 10 + 6 = 16.
3. Conclusion: For a non-degenerate triangle, x < 16 (x must be less than 16).

Triangle Inequality Theorem

• Definition: Any side of a triangle must be less than the sum of the other two sides if the triangle is non-degenerate.
• Application to Problem:
  • x must be less than 6 + 10 = 16.
  • To ensure non-degeneracy, x + 6 > 10 and 6 + 10 > x, leading to x > 4.

Summary

• Range for x: 4 < x < 16 (x is greater than 4 and less than 16).
• Key Principle: Triangle inequality theorem states a side must be less than the sum of the other two sides to form a non-degenerate triangle.
• Visualization: Helps in understanding extreme cases (minimizing and maximizing side length).