In `triangle ABC, AB = 6 cm, AC = 8 cm, and BC = 9 cm`. The length of median AD is :

Options

A. `(sqrt(317))/(2)cm`
B. `(sqrt(119))/(2)cm`
C. `(sqrt(115))/(2)cm`
D. `(sqrt(313))/(2)cm`

Answer

To find the length of the median AD in triangle ABC, where AB = 6 cm, AC = 8 cm, and BC = 9 cm, we can use the formula for the length of a median in a triangle. The median from vertex A to side BC is given by the formula:

\[ AD = \frac{1}{2} \sqrt{2AB^2 + 2AC^2 - BC^2} \]

### Step-by-Step Solution:

1. **Identify the lengths of the sides:**
- AB = 6 cm
- AC = 8 cm
- BC = 9 cm

2. **Substitute the values into the median formula:**
\[ AD = \frac{1}{2} \sqrt{2(6^2) + 2(8^2) - (9^2)} \]

3. **Calculate the squares of the sides:**
- \(6^2 = 36\)
- \(8^2 = 64\)
- \(9^2 = 81\)

4. **Substitute these values back into the equation:**
\[ AD = \frac{1}{2} \sqrt{2(36) + 2(64) - 81} \]

5. **Calculate the values inside the square root:**
- \(2(36) = 72\)
- \(2(64) = 128\)
- Therefore, \(AD = \frac{1}{2} \sqrt{72 + 128 - 81}\)

6. **Combine the terms:**
\[ AD = \frac{1}{2} \sqrt{119} \]

7. **Calculate the final value of AD:**
\[ AD = \frac{\sqrt{119}}{2} \]

### Final Answer:
The length of median AD is \(\frac{\sqrt{119}}{2}\) cm.