If P, R, T are the area of a Parallelogram, a rhombus and a triangle standing on the same base and between the same parallels, which of the following is true ?

Options

A. `R lt P lt T`
B. `P gt R gt T`
C. `R = P = T`
D. `R =P =2T`

Answer

To solve the problem, we need to analyze the areas of a parallelogram, a rhombus, and a triangle that share the same base and are between the same parallels. Let's denote the areas as follows:

- Let \( P \) be the area of the parallelogram.
- Let \( R \) be the area of the rhombus.
- Let \( T \) be the area of the triangle.

### Step-by-Step Solution:

1. **Understanding the Relationship Between Areas**:
- The area of a triangle that shares the same base and height as a parallelogram is half the area of the parallelogram. Therefore, we can express this relationship mathematically:
\[ T = \frac{1}{2} P \]

2. **Area of the Rhombus**:
- A rhombus can be considered a special type of parallelogram. Since the rhombus also shares the same base and height, its area is equal to that of the parallelogram:
\[ R = P \]

3. **Substituting the Area of the Rhombus**:
- From the first equation, we have \( T = \frac{1}{2} P \). From the second equation, we have \( R = P \). We can substitute \( R \) into the first equation:
\[ T = \frac{1}{2} R \]

4. **Expressing All Areas in Terms of \( T \)**:
- We can express \( P \) and \( R \) in terms of \( T \):
- From \( T = \frac{1}{2} P \), we can rearrange to find \( P \):
\[ P = 2T \]
- Since \( R = P \), we also have:
\[ R = 2T \]

5. **Final Relationships**:
- Now we can summarize the relationships:
\[ R = P = 2T \]

### Conclusion:
From our analysis, we conclude that:
\[ R = P = 2T \]