A point X inside a rectangle PQRS is joined to the vertices then, which of the following is true:

Options

A. `A(DeltaPSX)=A(DeltaRXQ)`
B. `A(DeltaPSX)+A(DeltaPXQ)=A(DeltaRSX)+A(DeltaRQX)`
C. `A(DeltaPXS)+A(DeltaRXQ)=A(DeltaSRX)+A(DeltaPXQ)`
D. none of these

Answer

To solve the problem, we need to analyze the areas of triangles formed by point X inside rectangle PQRS.

### Step-by-Step Solution

1. **Identify the Rectangle and Point**:
- We have a rectangle PQRS with vertices P, Q, R, and S.
- Let X be a point inside the rectangle.

2. **Understanding Triangle Areas**:
- When point X is connected to the vertices of the rectangle, it creates four triangles:
- Triangle PSX
- Triangle QSX
- Triangle RXQ
- Triangle PXQ

3. **Using the Area Relationship**:
- The key property we will use is that the area of triangles that share a common base and height can be compared.
- The area of triangle PSX is opposite to the area of triangle RXQ, and the area of triangle PXQ is opposite to the area of triangle QSX.

4. **Setting Up the Equation**:
- According to the properties of triangles formed by a point inside a rectangle, we can state:
\[ \text{Area of } \triangle PSX + \text{Area of } \triangle RXQ = \text{Area of } \triangle QSX + \text{Area of } \triangle PXQ \]

5. **Analyzing the Options**:
- Now, we can check the options given in the question:
- **Option 1**: Area of triangle PSX = Area of triangle RXQ (not true)
- **Option 2**: Area of triangle PSX + Area of triangle PXQ = Area of triangle RSX + Area of triangle RXQ (not true)
- **Option 3**: Area of triangle PSX + Area of triangle RXQ = Area of triangle RSX + Area of triangle PXQ (this is true)
- **Option 4**: None of these (not applicable)

6. **Conclusion**:
- The correct statement is that the area of triangle PSX plus the area of triangle RXQ is equal to the area of triangle QSX plus the area of triangle PXQ.

### Final Answer:
The correct option is **Option 3**: Area of triangle PSX + Area of triangle RXQ = Area of triangle QSX + Area of triangle PXQ.