In a triangle ABC if sides AB = c = 4 cm, side AC = b = 6 cm and BC = a = 7, then answer the following questions
The length of median AD is

Options

A. `53/2` cm
B. `1/2sqrt(55)` cm
C. `sqrt(53/2)` cm
D. `sqrt(63/2)` cm

Answer

To find the length of the median AD in triangle ABC where sides AB = c = 4 cm, AC = b = 6 cm, and BC = a = 7 cm, we can use the formula for the length of a median in a triangle.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- AB = c = 4 cm
- AC = b = 6 cm
- BC = a = 7 cm

2. **Use the median formula**:
The length of the median \( m_a \) from vertex A to side BC can be calculated using the formula:
\[ m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \]
where:
- \( a \) is the length of side BC,
- \( b \) is the length of side AC,
- \( c \) is the length of side AB.

3. **Substitute the values into the formula**:
\[ m_a = \frac{1}{2} \sqrt{2(6^2) + 2(4^2) - (7^2)} \]

4. **Calculate the squares**:
- \( 6^2 = 36 \)
- \( 4^2 = 16 \)
- \( 7^2 = 49 \)

5. **Plug in the squared values**:
\[ m_a = \frac{1}{2} \sqrt{2(36) + 2(16) - 49} \]
\[ = \frac{1}{2} \sqrt{72 + 32 - 49} \]

6. **Simplify inside the square root**:
\[ = \frac{1}{2} \sqrt{72 + 32 - 49} = \frac{1}{2} \sqrt{55} \]

7. **Final calculation**:
\[ = \frac{\sqrt{55}}{2} \]

Thus, the length of median AD is \( \frac{\sqrt{55}}{2} \) cm.