ABCD is a quadrilateral whose diagnonal AC divides it into two parts, equal in area, then ABCD

Options

A. is a rectangle
B. is always is rhombus
C. is a parallelogram
D. need not be any of (a), (b) or (c)

Explanation

To solve the problem, we need to determine the nature of the quadrilateral ABCD given that its diagonal AC divides it into two parts of equal area.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given a quadrilateral ABCD, and we know that the diagonal AC divides it into two triangles, ΔABC and ΔADC, which have equal areas.

2. **Area of Triangles**:
The area of triangle ΔABC is given by the formula:
\[ \text{Area}_{\Delta ABC} = \frac{1}{2} \times \text{base} \times \text{height} \]
Similarly, for triangle ΔADC:
\[ \text{Area}_{\Delta ADC} = \frac{1}{2} \times \text{base} \times \text{height} \]

3. **Equal Areas**:
Since the areas of the two triangles are equal, we can set up the equation:
\[ \text{Area}_{\Delta ABC} = \text{Area}_{\Delta ADC} \]
This implies:
\[ \frac{1}{2} \times \text{base}_{ABC} \times \text{height}_{ABC} = \frac{1}{2} \times \text{base}_{ADC} \times \text{height}_{ADC} \]

4. **Properties of Quadrilaterals**:
For a quadrilateral to have its diagonal dividing it into two equal areas, it can be a rectangle, rhombus, or parallelogram. All these shapes have the property that their diagonals bisect each other and divide the shape into two equal areas.

5. **Conclusion**:
Since the question states that the diagonal AC divides the quadrilateral ABCD into two equal areas, ABCD can be a rectangle, rhombus, or parallelogram. However, it does not necessarily have to be any of these shapes; it could also be any other quadrilateral that satisfies this condition.

6. **Final Answer**:
Therefore, the correct conclusion is that quadrilateral ABCD need not be any of the specified shapes (rectangle, rhombus, or parallelogram). Hence, the answer is option 4: "need not be any of A, B, and C".