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 le P le T`
B. `PgtRgtT`
C. R=P=T
D. R=P=2T

Answer

To solve the problem, we need to establish the relationships between the areas of the parallelogram (P), rhombus (R), and triangle (T) that share the same base and are between the same parallels.

### Step-by-Step Solution:

1. **Understanding the Shapes**:
- A parallelogram, rhombus, and triangle are all geometrical shapes that can be drawn on the same base and between the same parallels.

2. **Area of Triangle**:
- The area of a triangle is given by the formula:
\[ T = \frac{1}{2} \times \text{base} \times \text{height} \]
- Since the triangle shares the same base and height with the parallelogram, we can express the area of the triangle in terms of the area of the parallelogram.

3. **Area of Parallelogram**:
- The area of a parallelogram is given by the formula:
\[ P = \text{base} \times \text{height} \]

4. **Relationship Between Triangle and Parallelogram**:
- From the formulas above, we can see that:
\[ T = \frac{1}{2} P \]
- This means the area of the triangle is half the area of the parallelogram.

5. **Area of Rhombus**:
- A rhombus can also be considered a special case of a parallelogram. Therefore, if the base and height are the same, the area of the rhombus is equal to the area of the parallelogram:
\[ R = P \]

6. **Combining the Relationships**:
- We now have:
\[ T = \frac{1}{2} P \quad \text{and} \quad R = P \]
- From these equations, we can express T in terms of R:
\[ T = \frac{1}{2} R \]

7. **Final Relationships**:
- From the relationships established:
\[ P = R \quad \text{and} \quad T = \frac{1}{2} P = \frac{1}{2} R \]

8. **Conclusion**:
- Thus, we can conclude that:
\[ 2T = P = R \]
- The correct relationship among the areas is:
\[ \text{Area of Triangle} (T) = \frac{1}{2} \text{Area of Parallelogram} (P) = \frac{1}{2} \text{Area of Rhombus} (R) \]

### Final Answer:
The correct option is that \( P = R \) and \( T = \frac{1}{2} P \) or \( T = \frac{1}{2} R \).