Find length of AD in `Delta ABC ` if AB = 5 AC = 6 and BC = 8

Answer

To find the length of AD in triangle ABC where AB = 5, AC = 6, and BC = 8, we will use the formula for the length of a median. The median AD divides the opposite side BC into two equal parts, so we can denote BD = DC = 4 (since BC = 8).

### Step-by-step Solution:

1. **Identify the known values**:
- AB = 5
- AC = 6
- BC = 8
- Since AD is a median, BD = DC = BC/2 = 8/2 = 4.

2. **Use the median formula**:
The formula for the length of a median AD from vertex A to side BC is given by:
\[ AD^2 = \frac{2AB^2 + 2AC^2 - BC^2}{4} \]

3. **Substitute the known values into the formula**:
\[ AD^2 = \frac{2(5^2) + 2(6^2) - (8^2)}{4} \]

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

5. **Plug in the squared values**:
\[ AD^2 = \frac{2(25) + 2(36) - 64}{4} \]
\[ = \frac{50 + 72 - 64}{4} \]

6. **Simplify the expression**:
\[ = \frac{50 + 72 - 64}{4} = \frac{58}{4} = 14.5 \]

7. **Calculate AD**:
\[ AD = \sqrt{14.5} \approx 3.81 \]

### Final Result:
The length of AD is approximately \(3.81\).