A line PQ is drawn parallel to the side BC of `Delta ABC ` which cuts side AB at P and side AC at Q. If AB = 90 cm, CA = 60 cm and AQ = 4.2 cm, find the length of AP. 

Answer

To solve the problem step by step, 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 Information:**
- Length of AB = 90 cm
- Length of AC = 60 cm
- Length of AQ = 4.2 cm
- Let the length of AP = x (unknown)

2. **Set Up the Proportionality Equation:**
According to the Basic Proportionality Theorem, we have:
\[ \frac{AP}{AQ} = \frac{AB}{AC} \]
Substituting the known values:
\[ \frac{x}{4.2} = \frac{90}{60} \]

3. **Simplify the Right Side:**
The fraction \(\frac{90}{60}\) can be simplified:
\[ \frac{90}{60} = \frac{3}{2} \]
So, we rewrite the equation:
\[ \frac{x}{4.2} = \frac{3}{2} \]

4. **Cross Multiply to Solve for x:**
Cross multiplying gives us:
\[ 2x = 3 \times 4.2 \]
Calculate \(3 \times 4.2\):
\[ 3 \times 4.2 = 12.6 \]
Therefore, we have:
\[ 2x = 12.6 \]

5. **Divide by 2 to Find x:**
Now, divide both sides by 2:
\[ x = \frac{12.6}{2} = 6.3 \]

6. **Conclusion:**
The length of AP is:
\[ AP = 6.3 \text{ cm} \]

### Final Answer:
AP = 6.3 cm