In ∆DEW, AB || EW. If AD = 4 cm, DE = 12 cm and DW = 24 cm, then find the value of DB.

Answer

To solve the problem, we will use Thales' theorem, which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( AD = 4 \, \text{cm} \)
- \( DE = 12 \, \text{cm} \)
- \( DW = 24 \, \text{cm} \)
- Let \( DB = x \, \text{cm} \)
- Then, \( BW = DW - DB = 24 - x \, \text{cm} \)
- Let \( AE = DE - AD = 12 - 4 = 8 \, \text{cm} \)

2. **Apply Thales' Theorem:**
According to Thales' theorem:
\[ \frac{AD}{AE} = \frac{DB}{BW} \]
Substituting the known values:
\[ \frac{4}{8} = \frac{x}{24 - x} \]

3. **Cross-Multiply to Solve for \( x \):**
Cross-multiplying gives:
\[ 4(24 - x) = 8x \]
Expanding the left side:
\[ 96 - 4x = 8x \]

4. **Combine Like Terms:**
Adding \( 4x \) to both sides:
\[ 96 = 8x + 4x \]
\[ 96 = 12x \]

5. **Solve for \( x \):**
Dividing both sides by 12:
\[ x = \frac{96}{12} = 8 \, \text{cm} \]

6. **Conclusion:**
Therefore, the value of \( DB \) is \( 8 \, \text{cm} \).

### Final Answer:
\( DB = 8 \, \text{cm} \)