Which of the following represents the sides of an acute angled triangle?

Options

A. 6, 9, 16
B. 7, 8, 10
C. 5, 12, 13
D. None of these

Answer

To determine which of the given options represents the sides of an acute-angled triangle, we need to apply the properties of acute-angled triangles. An acute-angled triangle has all angles less than 90 degrees, which also implies that the square of the length of the longest side must be less than the sum of the squares of the lengths of the other two sides.

### Step-by-Step Solution:

1. **Identify the Properties of an Acute-Angled Triangle:**
- In an acute-angled triangle, if we denote the sides as \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the following condition must hold:
\[ c^2 < a^2 + b^2 \]

2. **Evaluate Each Option:**
- We will check each option to see if it satisfies the condition for being the sides of an acute-angled triangle.

3. **Option A: (16, 9, 6)**
- Identify the longest side: \( c = 16 \)
- Check the condition:
\[ 16^2 < 9^2 + 6^2 \]
\[ 256 < 81 + 36 \]
\[ 256 < 117 \quad \text{(False)} \]
- Therefore, Option A does not represent the sides of an acute-angled triangle.

4. **Option B: (10, 7, 8)**
- Identify the longest side: \( c = 10 \)
- Check the condition:
\[ 10^2 < 7^2 + 8^2 \]
\[ 100 < 49 + 64 \]
\[ 100 < 113 \quad \text{(True)} \]
- Therefore, Option B satisfies the condition and represents the sides of an acute-angled triangle.

5. **Conclusion:**
- The correct option that represents the sides of an acute-angled triangle is **Option B: (10, 7, 8)**.