If perimeter of rectangle & Parallelogram is equal with same base. Find the ratio of area of rectangle and area of parallelogram.

Options

A. `=1`
B. `gt1`
C. `lt1`
D. None of these

Answer

To solve the problem, we need to find the ratio of the area of a rectangle to the area of a parallelogram given that their perimeters are equal and they share the same base.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the base of both the rectangle and the parallelogram be \( L \).
- Let the height of the rectangle be \( B \).
- Let the height of the parallelogram be \( h \).

2. **Calculate the Perimeter**:
- The perimeter of the rectangle is given by:
\[ P_{rectangle} = 2(L + B) \]
- The perimeter of the parallelogram is given by:
\[ P_{parallelogram} = 2(L + h) \]
- Since the perimeters are equal:
\[ 2(L + B) = 2(L + h) \]
- Dividing both sides by 2:
\[ L + B = L + h \]
- This simplifies to:
\[ B = h \]

3. **Calculate the Areas**:
- The area of the rectangle is:
\[ A_{rectangle} = L \times B \]
- The area of the parallelogram is:
\[ A_{parallelogram} = L \times h \]
- Since we established that \( B = h \), we can substitute \( h \) with \( B \):
\[ A_{parallelogram} = L \times B \]

4. **Find the Ratio of Areas**:
- Now, we can find the ratio of the area of the rectangle to the area of the parallelogram:
\[ \text{Ratio} = \frac{A_{rectangle}}{A_{parallelogram}} = \frac{L \times B}{L \times h} \]
- Since \( B = h \):
\[ \text{Ratio} = \frac{L \times B}{L \times B} = 1 \]

5. **Conclusion**:
- The ratio of the area of the rectangle to the area of the parallelogram is:
\[ \text{Ratio} = 1 \]