In a triangle ABC if sides AB = c = 4 cm, side AC = b = 6 cm and BC = a = 7, then answer the following questions
If AD be the altitude then what is BD in cm ?

Options

A. `29/12`
B. `39/12`
C. `29/14`
D. `69/14`

Answer

To find the length of segment BD in triangle ABC, where AD is the altitude from A to side BC, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- AB (c) = 4 cm
- AC (b) = 6 cm
- BC (a) = 7 cm

2. **Draw the Triangle:**
- Draw triangle ABC with points A, B, and C.
- Mark the sides as given: AB = 4 cm, AC = 6 cm, and BC = 7 cm.
- Draw the altitude AD from point A to side BC, meeting BC at point D.

3. **Define BD and DC:**
- Let BD = y cm.
- Therefore, DC = BC - BD = 7 - y cm.

4. **Apply the Pythagorean Theorem:**
- In triangle ABD:
\[ AD^2 + BD^2 = AB^2 \]
\[ AD^2 + y^2 = 4^2 \quad (1) \]
- In triangle ACD:
\[ AD^2 + DC^2 = AC^2 \]
\[ AD^2 + (7 - y)^2 = 6^2 \quad (2) \]

5. **Set Up the Equations:**
- From equation (1):
\[ AD^2 = 16 - y^2 \quad (3) \]
- From equation (2):
\[ AD^2 = 36 - (7 - y)^2 \quad (4) \]

6. **Expand and Equate the Two Expressions for AD²:**
- Expand equation (4):
\[ AD^2 = 36 - (49 - 14y + y^2) \]
\[ AD^2 = 36 - 49 + 14y - y^2 \]
\[ AD^2 = -13 + 14y - y^2 \quad (5) \]

7. **Set Equations (3) and (5) Equal:**
- From (3) and (5):
\[ 16 - y^2 = -13 + 14y - y^2 \]
- Cancel out -y² from both sides:
\[ 16 = -13 + 14y \]
- Rearranging gives:
\[ 14y = 16 + 13 \]
\[ 14y = 29 \]
\[ y = \frac{29}{14} \]

8. **Conclusion:**
- Therefore, BD = y = \(\frac{29}{14}\) cm.

### Final Answer:
BD = \(\frac{29}{14}\) cm.