Each side of a square ABCD is 12 cm A point P l lies on side DC such that area of ` Delta ADP ` : area of trapezium ABCP = 2 : 3 Find DP .

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a square ABCD with each side measuring 12 cm. A point P lies on side DC. We need to find the length of DP given that the ratio of the area of triangle ADP to the area of trapezium ABCP is 2:3.

2. **Identify the Areas**:
- The area of triangle ADP can be calculated using the formula for the area of a triangle:
\[ \text{Area of } \Delta ADP = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is DP and the height is AD (which is 12 cm).
\[ \text{Area of } \Delta ADP = \frac{1}{2} \times DP \times 12 = 6 \times DP \]

3. **Calculate the Area of Trapezium ABCP**:
- The area of trapezium ABCP can be calculated using the formula:
\[ \text{Area of trapezium} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} \]
The parallel sides are AB (12 cm) and PC. The height is BC (12 cm).
\[ \text{Area of trapezium ABCP} = \frac{1}{2} \times (12 + PC) \times 12 = 6 \times (12 + PC) \]

4. **Set Up the Ratio**:
- According to the problem, the ratio of the areas is given as:
\[ \frac{\text{Area of } \Delta ADP}{\text{Area of trapezium ABCP}} = \frac{2}{3} \]
- Substituting the areas we calculated:
\[ \frac{6 \times DP}{6 \times (12 + PC)} = \frac{2}{3} \]
- Simplifying gives:
\[ \frac{DP}{12 + PC} = \frac{2}{3} \]

5. **Cross Multiply**:
- Cross multiplying gives:
\[ 3 \times DP = 2 \times (12 + PC) \]
- Expanding this:
\[ 3DP = 24 + 2PC \]

6. **Relate DP and PC**:
- Since P lies on DC, we know that:
\[ DP + PC = 12 \]
- Rearranging gives:
\[ PC = 12 - DP \]

7. **Substituting PC**:
- Substitute PC in the equation from step 5:
\[ 3DP = 24 + 2(12 - DP) \]
- Simplifying gives:
\[ 3DP = 24 + 24 - 2DP \]
\[ 3DP + 2DP = 48 \]
\[ 5DP = 48 \]
\[ DP = \frac{48}{5} = 9.6 \text{ cm} \]

### Final Answer:
The length of DP is **9.6 cm**.