In a `Delta ABC, AB=5, AC=6cm, BC=8 cm` . Find the length of median AD.

Answer

To find the length of the median AD in triangle ABC where AB = 5 cm, AC = 6 cm, and BC = 8 cm, we can use the formula for the length of the median from vertex A to side BC.

### Step-by-Step Solution:

1. **Identify the lengths of the sides:**
- Let AB = c = 5 cm
- Let AC = b = 6 cm
- Let BC = a = 8 cm

2. **Determine the length of the median AD:**
The formula for the length of the median (m_a) from vertex A to side BC is given by:
\[ m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \]
where:
- \( b \) is the length of side AC,
- \( c \) is the length of side AB,
- \( a \) is the length of side BC.

3. **Substitute the values into the formula:**
\[ m_a = \frac{1}{2} \sqrt{2(6^2) + 2(5^2) - (8^2)} \]

4. **Calculate the squares of the sides:**
- \( 6^2 = 36 \)
- \( 5^2 = 25 \)
- \( 8^2 = 64 \)

5. **Plug the squared values into the formula:**
\[ m_a = \frac{1}{2} \sqrt{2(36) + 2(25) - 64} \]
\[ = \frac{1}{2} \sqrt{72 + 50 - 64} \]
\[ = \frac{1}{2} \sqrt{58} \]

6. **Calculate the median:**
\[ m_a = \frac{1}{2} \sqrt{58} \approx \frac{1}{2} \times 7.62 \approx 3.81 \text{ cm} \]

### Final Answer:
The length of the median AD is approximately 3.81 cm.