In `DeltaABC, DEabs()BC`. If AD= 4 cm, DB = 4.5cm and AE = 8cm, find AC.

Answer

To solve the problem, we will use the properties of similar triangles. Since DE is parallel to BC, we can use the concept of proportionality in triangles.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- AD = 4 cm
- DB = 4.5 cm
- AE = 8 cm
- EC = ? (unknown)
- AC = AE + EC (we need to find this)

2. **Set Up the Proportionality:**
Since DE is parallel to BC, we can use the property that states:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]

3. **Substitute the Known Values:**
Substitute the known lengths into the proportion:
\[ \frac{4}{4.5} = \frac{8}{EC} \]

4. **Cross-Multiply to Solve for EC:**
Cross-multiplying gives us:
\[ 4 \cdot EC = 8 \cdot 4.5 \]
\[ 4 \cdot EC = 36 \]

5. **Divide to Find EC:**
Now, divide both sides by 4:
\[ EC = \frac{36}{4} = 9 \text{ cm} \]

6. **Calculate AC:**
Now that we have EC, we can find AC:
\[ AC = AE + EC = 8 \text{ cm} + 9 \text{ cm} = 17 \text{ cm} \]

### Final Answer:
The length of AC is **17 cm**.