IC Which sets of three of numbers represent the sides of an obtuse triangle? Check all that apply \( 4,7,8 \) \( 3,4,5 \) 2, 2,3 \( 6,8,9 \) \( 3,5,6 \)

To determine which sets of three numbers can form the sides of an **obtuse triangle**, we'll follow these steps: 1. **Check the Triangle Inequality Theorem**: - The sum of any two sides must be greater than the third side. 2. **Determine the Type of Triangle**: - For sides \(a \leq b \leq c\): - **Obtuse Triangle**: \(a^2 + b^2 < c^2\) - **Right Triangle**: \(a^2 + b^2 = c^2\) - **Acute Triangle**: \(a^2 + b^2 > c^2\) Let's evaluate each set: 1. **\(4, 7, 8\)** - **Triangle Inequality**: \(4 + 7 > 8\), \(4 + 8 > 7\), \(7 + 8 > 4\) → **Valid Triangle** - **Type**: \[ 4^2 + 7^2 = 16 + 49 = 65 \\ 8^2 = 64 \\ 65 > 64 \Rightarrow \text{Acute Triangle} \] - **Result**: **Not Obtuse** 2. **\(3, 4, 5\)** - **Triangle Inequality**: \(3 + 4 > 5\), \(3 + 5 > 4\), \(4 + 5 > 3\) → **Valid Triangle** - **Type**: \[ 3^2 + 4^2 = 9 + 16 = 25 \\ 5^2 = 25 \\ 25 = 25 \Rightarrow \text{Right Triangle} \] - **Result**: **Not Obtuse** 3. **\(2, 2, 3\)** - **Triangle Inequality**: \(2 + 2 > 3\), \(2 + 3 > 2\), \(2 + 3 > 2\) → **Valid Triangle** - **Type**: \[ 2^2 + 2^2 = 4 + 4 = 8 \\ 3^2 = 9 \\ 8 < 9 \Rightarrow \text{Obtuse Triangle} \] - **Result**: **Obtuse** 4. **\(6, 8, 9\)** - **Triangle Inequality**: \(6 + 8 > 9\), \(6 + 9 > 8\), \(8 + 9 > 6\) → **Valid Triangle** - **Type**: \[ 6^2 + 8^2 = 36 + 64 = 100 \\ 9^2 = 81 \\ 100 > 81 \Rightarrow \text{Acute Triangle} \] - **Result**: **Not Obtuse** 5. **\(3, 5, 6\)** - **Triangle Inequality**: \(3 + 5 > 6\), \(3 + 6 > 5\), \(5 + 6 > 3\) → **Valid Triangle** - **Type**: \[ 3^2 + 5^2 = 9 + 25 = 34 \\ 6^2 = 36 \\ 34 < 36 \Rightarrow \text{Obtuse Triangle} \] - **Result**: **Obtuse** ### **Conclusion** The following sets represent the sides of an **obtuse triangle**: - \( \mathbf{2, 2, 3} \) - \( \mathbf{3, 5, 6} \) **Hence, the correct options are:** - \(2, 2, 3\) - \(3, 5, 6\)