ABCDis a rhombus with diagonalsACandBD.Then, which one among the following is correct?

Options

A. AC and BD bisect each other but not necessarily perpendicular to each other.
B. AC and BD are perpendicular to each other but not necessarily bisect each other.
C. AC and BD bisect each other and perpendicular to each other.
D. AC and BD neither bisect each other nor perpendicular to each other

Answer

To solve the question regarding the properties of the diagonals of a rhombus ABCD, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Properties of a Rhombus**:
- A rhombus is a type of quadrilateral where all four sides are of equal length.
- The diagonals of a rhombus have specific properties: they bisect each other and are perpendicular to each other.

2. **Identifying the Diagonals**:
- In rhombus ABCD, the diagonals are AC and BD.
- By definition, the point where the diagonals intersect is the midpoint of both diagonals.

3. **Analyzing the Options**:
- **Option A**: "AC and BD bisect each other but not necessarily perpendicular."
- This is incorrect because, in a rhombus, the diagonals not only bisect each other but also intersect at right angles (90 degrees).

- **Option B**: "AC and BD are perpendicular but not necessarily bisect."
- This is also incorrect because the diagonals do bisect each other in a rhombus.

- **Option C**: "AC and BD bisect each other and are perpendicular to each other."
- This is correct as it accurately describes the properties of the diagonals in a rhombus.

- **Option D**: "AC and BD neither bisect nor are perpendicular."
- This is incorrect because it contradicts the properties of the diagonals in a rhombus.

4. **Conclusion**:
- The correct option is **Option C**: "AC and BD bisect each other and are perpendicular to each other."