ABCD is a square with each side 12 cm. P is a point on BC such that area of `DeltaABP` : area of trapezium APCD = `1 : 5`. Find the length of CP.

Answer

To solve the problem, we need to find the length of CP given that ABCD is a square with each side measuring 12 cm, and the ratio of the area of triangle ABP to the area of trapezium APCD is 1:5.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a square ABCD with vertices A(0, 12), B(0, 0), C(12, 0), and D(12, 12).
- Point P is on side BC, which means its coordinates can be expressed as P(0, y) where 0 ≤ y ≤ 12.

2. **Identify Areas**:
- The area of triangle ABP can be calculated using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base AB is 12 cm (the height from A to line BP is the y-coordinate of P).

3. **Calculate Area of Triangle ABP**:
- The base AB = 12 cm and height = y (the y-coordinate of point P).
- Thus, the area of triangle ABP is:
\[ \text{Area}_{ABP} = \frac{1}{2} \times 12 \times y = 6y \]

4. **Calculate Area of Square ABCD**:
- The area of square ABCD is:
\[ \text{Area}_{ABCD} = 12 \times 12 = 144 \text{ cm}^2 \]

5. **Calculate Area of Trapezium APCD**:
- The area of trapezium APCD can be found by subtracting the area of triangle ABP from the area of square ABCD:
\[ \text{Area}_{APCD} = \text{Area}_{ABCD} - \text{Area}_{ABP} = 144 - 6y \]

6. **Set Up the Ratio**:
- According to the problem, the ratio of the area of triangle ABP to the area of trapezium APCD is 1:5:
\[ \frac{6y}{144 - 6y} = \frac{1}{5} \]

7. **Cross Multiply to Solve for y**:
- Cross multiplying gives:
\[ 5 \times 6y = 1 \times (144 - 6y) \]
- This simplifies to:
\[ 30y = 144 - 6y \]
- Adding 6y to both sides results in:
\[ 36y = 144 \]
- Dividing both sides by 36 gives:
\[ y = 4 \]

8. **Find Length of CP**:
- Since P is on BC, and BC is vertical, the length CP is the distance from C(12, 0) to P(0, 4).
- The length CP can be calculated as:
\[ CP = 12 - y = 12 - 4 = 8 \text{ cm} \]

### Final Answer:
The length of CP is **8 cm**.