How to Solve: Triangle Inequality

How to Solve: Triangle Inequality

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you’ve ever wondered why a 3-inch, 4-inch, and 10-inch stick can’t form a triangle—this is the rule that explains it. Master the Triangle Inequality, and you’ll solve geometry problems faster, avoid traps, and even impress your teacher on exam day."

What You Need To Know First

Before diving in, make sure you understand: 1. What a triangle is – A 3-sided polygon with 3 angles. 2. How to measure side lengths – Use a ruler or given values (e.g., AB = 5 cm). 3. Basic inequality symbols – < (less than), > (greater than), ≤ (less than or equal to).

Key Vocabulary

Formulas To Know

1. Triangle Inequality Theorem (MEMORISE THIS)

For any triangle with sides a, b, and c: - a + b > c - a + c > b - b + c > a

What the variables mean: - a, b, c = lengths of the 3 sides of the triangle.

Why it matters: If any of these inequalities is false, the sides cannot form a triangle.

Step-by-Step Method

Follow these steps exactly for every problem.

1. Label the sides – Write down the 3 side lengths (e.g., a = 5, b = 7, c = 10).
2. Write all 3 inequalities – Use the Triangle Inequality Theorem:
3. a + b > c
4. a + c > b
5. b + c > a
6. Plug in the numbers – Replace a, b, and c with the given lengths.
7. Calculate each sum – Add the two sides in each inequality.
8. Check if the inequality is true – If the sum is greater than the third side, the inequality holds.
9. Count the true inequalities – If all 3 are true, the sides can form a triangle. If even one is false, they cannot.

Worked Example Using the Steps

Problem: Can sides of lengths 6, 8, and 14 form a triangle?

1. Label the sides:
2. a = 6
3. b = 8

4. c = 14

5. Write all 3 inequalities:

6. 6 + 8 > 14
7. 6 + 14 > 8

8. 8 + 14 > 6

9. Plug in the numbers:

10. 14 > 14 (Wait, 6 + 8 = 14, not greater than 14!)
11. 20 > 8 (True)

12. 22 > 6 (True)

13. Check each inequality:

14. 14 > 14 → False (14 is not greater than 14)
15. 20 > 8 → True

16. 22 > 6 → True

17. Count the true inequalities:

18. Only 2 out of 3 are true.

19. Final Answer:

20. No, these sides cannot form a triangle because one inequality fails.

Worked Examples

Example 1 – Basic (No Tricks)

Problem: Can sides of lengths 3, 4, and 5 form a triangle?

Solution: 1. Label: a = 3, b = 4, c = 5 2. Write inequalities:
- 3 + 4 > 5 → 7 > 5 (True)
- 3 + 5 > 4 → 8 > 4 (True)
- 4 + 5 > 3 → 9 > 3 (True) 3. All 3 inequalities are true.

Answer: Yes, these sides can form a triangle.

What we did and why: We checked all 3 possible sums. Since every sum was greater than the third side, the sides satisfy the Triangle Inequality Theorem.

Example 2 – Medium (One Complication)

Problem: A triangle has sides of lengths 7 and 10. What is the range of possible lengths for the third side?

Solution: 1. Let the third side be x. 2. Write the 3 inequalities:
- 7 + 10 > x → 17 > x → x < 17
- 7 + x > 10 → x > 3
- 10 + x > 7 → x > -3 (This is always true since side lengths are positive.) 3. Combine the useful inequalities:
- x < 17 and x > 3 4. Write the range:
- 3 < x < 17

Answer: The third side must be greater than 3 and less than 17.

What we did and why: We used the Triangle Inequality to find the minimum (x > 3) and maximum (x < 17) possible lengths for the third side.

Example 3 – Exam Style (Disguised Problem)

Problem: A student draws a triangle with sides 2x, 5x, and 4x + 3. For what values of x is this triangle possible?

Solution: 1. Write the 3 inequalities:
- 2x + 5x > 4x + 3 → 7x > 4x + 3 → 3x > 3 → x > 1
- 2x + (4x + 3) > 5x → 6x + 3 > 5x → x > -3 (Always true since x must be positive.)
- 5x + (4x + 3) > 2x → 9x + 3 > 2x → 7x > -3 (Always true.) 2. The only restrictive inequality is x > 1. 3. Since side lengths must be positive:
- 2x > 0 → x > 0
- 5x > 0 → x > 0
- 4x + 3 > 0 → x > -0.75 (Always true if x > 0.)

Answer: The triangle is possible when x > 1.

What we did and why: We treated x as a variable and solved the inequalities to find the minimum value of x that makes the triangle possible.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—especially if your exam is tomorrow! The Triangle Inequality Theorem says: For any triangle, the sum of any two sides must be greater than the third side. That means you always check three inequalities. If all three are true, the sides can form a triangle. If even one fails, they can’t. Watch out for traps like disguised side lengths or range questions—just follow the steps, and you’ll nail it. Now go practice a few problems, and you’ll be ready to crush this on exam day!