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 BD : DC ?

Options

A. `29:69`
B. `69:29`
C. `29:39`
D. `39:29`

Answer

To solve the problem, we need to find the ratio \( BD : DC \) in triangle \( ABC \) where \( AB = 4 \, \text{cm} \), \( AC = 6 \, \text{cm} \), and \( BC = 7 \, \text{cm} \). We will denote \( BD = y \) and \( DC = x \).

### Step-by-step Solution:

1. **Draw Triangle ABC**:
- Draw triangle \( ABC \) with sides \( AB = c = 4 \, \text{cm} \), \( AC = b = 6 \, \text{cm} \), and \( BC = a = 7 \, \text{cm} \).
- Mark the points \( A, B, \) and \( C \) accordingly.

**Hint**: Visualizing the triangle helps in understanding the relationships between the sides and the segments.

2. **Draw the Altitude AD**:
- Draw the altitude \( AD \) from point \( A \) to line \( BC \), meeting at point \( D \).
- This creates two segments on line \( BC \): \( BD \) and \( DC \).

**Hint**: The altitude helps in applying the Pythagorean theorem to find the lengths of segments.

3. **Define the Segments**:
- Let \( DC = x \).
- Therefore, \( BD = 7 - x \) (since the total length \( BC = 7 \, \text{cm} \)).

**Hint**: Expressing segments in terms of a single variable simplifies calculations.

4. **Apply the Pythagorean Theorem**:
- In triangle \( ADC \):
\[ AD^2 + DC^2 = AC^2 \implies y^2 + x^2 = 6^2 \implies y^2 + x^2 = 36 \quad \text{(1)} \]
- In triangle \( ADB \):
\[ AD^2 + BD^2 = AB^2 \implies y^2 + (7 - x)^2 = 4^2 \implies y^2 + (7 - x)^2 = 16 \quad \text{(2)} \]

**Hint**: The Pythagorean theorem is crucial for relating the sides of right triangles.

5. **Expand Equation (2)**:
- Expanding \( (7 - x)^2 \):
\[ y^2 + (49 - 14x + x^2) = 16 \]
- Rearranging gives:
\[ y^2 + x^2 - 14x + 49 = 16 \implies y^2 + x^2 - 14x + 33 = 0 \quad \text{(3)} \]

**Hint**: Rearranging and combining like terms helps in isolating variables.

6. **Substitute Equation (1) into Equation (3)**:
- From (1), \( y^2 = 36 - x^2 \).
- Substitute into (3):
\[ (36 - x^2) + x^2 - 14x + 33 = 0 \]
- This simplifies to:
\[ 69 - 14x = 0 \implies 14x = 69 \implies x = \frac{69}{14} \]

**Hint**: Substituting known values can simplify the equation significantly.

7. **Calculate \( BD \)**:
- Now, \( BD = 7 - x = 7 - \frac{69}{14} = \frac{98}{14} - \frac{69}{14} = \frac{29}{14} \).

**Hint**: Ensure to keep track of the fractions correctly.

8. **Find the Ratio \( BD : DC \)**:
- The ratio \( BD : DC \) is:
\[ \frac{BD}{DC} = \frac{\frac{29}{14}}{\frac{69}{14}} = \frac{29}{69} \]

**Hint**: Ratios can often be simplified by canceling out common factors.

### Final Answer:
The ratio \( BD : DC = 29 : 69 \).