If in `triangleABC`, AD is the bisector of `angleA and D` lies on BC. If AB = 6.4 cm, AC = 8 cm, BD = 5.6 cm, find DC.

Answer

To solve the problem, we will use the Angle Bisector Theorem, which states that the ratio of the two segments created by the angle bisector on the opposite side is equal to the ratio of the other two sides of the triangle.

### Step-by-Step Solution:

1. **Identify Given Values:**
- \( AB = 6.4 \, \text{cm} \)
- \( AC = 8 \, \text{cm} \)
- \( BD = 5.6 \, \text{cm} \)
- Let \( DC = x \, \text{cm} \)

2. **Apply the Angle Bisector Theorem:**
According to the Angle Bisector Theorem:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]
Substituting the known values:
\[ \frac{6.4}{8} = \frac{5.6}{x} \]

3. **Cross Multiply:**
Cross multiplying gives us:
\[ 6.4 \cdot x = 8 \cdot 5.6 \]

4. **Calculate \( 8 \cdot 5.6 \):**
\[ 8 \cdot 5.6 = 44.8 \]
So, we have:
\[ 6.4x = 44.8 \]

5. **Solve for \( x \):**
Divide both sides by \( 6.4 \):
\[ x = \frac{44.8}{6.4} \]

6. **Calculate \( \frac{44.8}{6.4} \):**
\[ x = 7 \, \text{cm} \]

7. **Conclusion:**
Therefore, \( DC = 7 \, \text{cm} \).

### Final Answer:
\( DC = 7 \, \text{cm} \)