Two sides of a triangle measure 4 cm and 7 cm. Which one of the following CANNOT be a measure of the
third side?
(1) 4 cm
(2) 5 cm
(3) 8 cm
(4) 11 cm

Triangle Inequality Rule: Which Side Is NOT Possible?

Geometry problems often test our understanding of basic theorems — and one of the most fundamental is the Triangle Inequality Theorem. Let’s break down a classic question:

Two sides of a triangle measure 4 cm and 7 cm. Which one of the following cannot be the third side?

Options:

1. 4 cm

2. 5 cm

3. 8 cm

4. 11 cm

🧠 Triangle Inequality Theorem — Quick Refresher

The Triangle Inequality Theorem states that:

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

So, for three sides A, B, and C:

• A + B > C

• B + C > A

• C + A > B

Let’s apply this to the problem.

🔍 Given:

• Side A = 4 cm

• Side B = 7 cm

• Let Side C = unknown (we’re checking all 4 options)

Let’s test each option as a possible third side.

✅ Option 1: 4 cm

Check:
4 + 4 = 8 > 7 ✅
4 + 7 = 11 > 4 ✅
7 + 4 = 11 > 4 ✅
Valid

✅ Option 2: 5 cm

4 + 5 = 9 > 7 ✅
5 + 7 = 12 > 4 ✅
4 + 7 = 11 > 5 ✅
Valid

✅ Option 3: 8 cm

4 + 8 = 12 > 7 ✅
7 + 8 = 15 > 4 ✅
4 + 7 = 11 > 8 ✅
Valid

❌ Option 4: 11 cm

Check: 4 + 7 = 11
But 11 is NOT greater than 11 → ❌ violates the triangle inequality.

So, a triangle cannot have sides 4 cm, 7 cm, and 11 cm.

✅ Correct Answer: (4) 11 cm

🔑 Key Takeaway

If you’re ever unsure whether three side lengths can form a triangle, just apply the Triangle Inequality Theorem:

Sum of any two sides must be greater than the third.

📘 Why This Matters

Understanding triangle properties is essential for:

• Geometry and math exams

• Engineering and architecture

• Problem-solving in physics and design

Final Thoughts

The third side of a triangle with 4 cm and 7 cm cannot be 11 cm, because it violates the triangle inequality. Always test side combinations to verify that they meet the condition:

Sum of two sides > the third