P and Q are the points on sides AB and AC respectively of a `DeltaABC` , such that PQ||BC .If `AP|PB=2:3` and AQ=4 cm , then the length of AC is ….. Cm .

Answer

To solve the problem, we will use the Basic Proportionality Theorem (also known as 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 Ratios and Lengths:**
- We are given that \( \frac{AP}{PB} = \frac{2}{3} \).
- The length of \( AQ \) is given as \( 4 \) cm.

2. **Set Up the Proportionality:**
- According to the Basic Proportionality Theorem, since \( PQ \parallel BC \), we have:
\[ \frac{AP}{PB} = \frac{AQ}{QC} \]
- Let \( QC \) be \( x \) cm. Therefore, we can write:
\[ \frac{2}{3} = \frac{4}{x} \]

3. **Cross-Multiply to Solve for \( x \):**
- Cross-multiplying gives:
\[ 2x = 3 \times 4 \]
- Simplifying this, we find:
\[ 2x = 12 \implies x = \frac{12}{2} = 6 \text{ cm} \]
- Thus, \( QC = 6 \) cm.

4. **Calculate the Length of AC:**
- Now, we can find the total length of \( AC \):
\[ AC = AQ + QC = 4 \text{ cm} + 6 \text{ cm} = 10 \text{ cm} \]

5. **Final Answer:**
- Therefore, the length of \( AC \) is \( 10 \) cm.