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

### Step-by-Step Solution:

1. **Identify the given values:**
- AB = 6.4 cm
- AC = 8 cm
- BD = 5.6 cm
- Let DC = x (we need to find this value).

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

3. **Cross-multiply to eliminate the fraction:**
\[ 5.6 \cdot 8 = 6.4 \cdot x \]
This simplifies to:
\[ 44.8 = 6.4x \]

4. **Solve for x:**
To find x, divide both sides by 6.4:
\[ x = \frac{44.8}{6.4} \]

5. **Calculate the value:**
\[ x = 7 \]
Thus, DC = 7 cm.

### Final Answer:
DC = 7 cm.