ABCD is a parallelogram P and R are two points on AB such that the area of parallelogram ABCD is 8 times the area of `Delta` DPR. If PR = 5 cm, then CD is equal to

Options

A. 10 cm
B. 5 cm
C. 20 cm
D. 12 cm

Answer

To solve the problem step by step, we need to find the length of CD in the parallelogram ABCD given that the area of parallelogram ABCD is 8 times the area of triangle DPR and that PR = 5 cm.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- ABCD is a parallelogram.
- P and R are points on AB.
- Area of parallelogram ABCD = 8 × Area of triangle DPR.
- Length of PR = 5 cm.

2. **Area of Parallelogram**:
- The formula for the area of a parallelogram is:
\[ \text{Area of parallelogram} = \text{Base} \times \text{Height} \]
- Let the base be AB and the height be h. Thus:
\[ \text{Area of parallelogram ABCD} = AB \times h \]

3. **Area of Triangle DPR**:
- The formula for the area of a triangle is:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- Here, the base is PR (which is 5 cm) and the height is also h (same height as the parallelogram). Thus:
\[ \text{Area of triangle DPR} = \frac{1}{2} \times PR \times h = \frac{1}{2} \times 5 \times h = \frac{5h}{2} \]

4. **Set Up the Equation**:
- According to the problem, the area of parallelogram ABCD is 8 times the area of triangle DPR:
\[ AB \times h = 8 \times \left(\frac{5h}{2}\right) \]

5. **Simplify the Equation**:
- Simplifying the right side:
\[ AB \times h = 8 \times \frac{5h}{2} = 20h \]
- Now, we can cancel h from both sides (assuming h ≠ 0):
\[ AB = 20 \text{ cm} \]

6. **Find CD**:
- In a parallelogram, opposite sides are equal. Therefore, CD is equal to AB:
\[ CD = AB = 20 \text{ cm} \]

### Final Answer:
CD = 20 cm.