21. Which of the sets of numbers can be the lengths of a triangle? 2.6,8.1,10.2 $$ 6,6,12 $$ $$ 4.3,5.7,12.1 $$ $$ 6,7,13 $$

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To determine which sets of numbers can represent the lengths of a triangle, we apply the **Triangle Inequality Theorem**. This theorem states that for any three lengths to form a triangle, the sum of any two sides must be **greater** than the third side.

Let's evaluate each set:

1. **2.6, 8.1, 10.2**
   - **2.6 + 8.1 = 10.7 > 10.2** ✅
   - **2.6 + 10.2 = 12.8 > 8.1** ✅
   - **8.1 + 10.2 = 18.3 > 2.6** ✅
   
   **Conclusion:** These lengths satisfy the triangle inequality and can form a triangle.

2. **6, 6, 12**
   - **6 + 6 = 12 = 12** ❌
   
   **Conclusion:** The sum of two sides equals the third side, which does not satisfy the strict inequality required. Therefore, these lengths **cannot** form a triangle.

3. **4.3, 5.7, 12.1**
   - **4.3 + 5.7 = 10 < 12.1** ❌
   
   **Conclusion:** The sum of the two smaller sides is less than the third side. These lengths **cannot** form a triangle.

4. **6, 7, 13**
   - **6 + 7 = 13 = 13** ❌
   
   **Conclusion:** Similar to the second set, the sum of two sides equals the third side, violating the triangle inequality. These lengths **cannot** form a triangle.

**Final Answer:** Only the set **2.6, 8.1, 10.2** can be the lengths of a triangle.
Only the set **2.6, 8.1, 10.2** can form a triangle.

21. Which of the sets of numbers can be the lengths of a triangle? 2.6,8.1,10.2 \( 6,6,12 \) \( 4.3,5.7,12.1 \) \( 6,7,13 \)

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