If two diameters of a circle intersect each other at right angles, then quadrilateral formed by joining their end points is a

Options

A. Rhombus
B. rectangle
C. square
D. parallelogram

Answer

To solve the problem, we need to analyze the situation where two diameters of a circle intersect at right angles and determine the nature of the quadrilateral formed by joining their endpoints.

### Step-by-step Solution:

1. **Understanding the Circle and Diameters**:
- Let the circle have a center O.
- Let the two diameters be AC and BD, which intersect at point O at right angles (90 degrees).

2. **Identifying the Endpoints**:
- The endpoints of diameter AC are points A and C.
- The endpoints of diameter BD are points B and D.
- Thus, the quadrilateral formed by joining these endpoints is ABCD.

3. **Analyzing the Angles**:
- Since AC and BD are diameters and they intersect at right angles, we can conclude that:
- Angle AOB = 90 degrees
- Angle BOC = 90 degrees
- Angle COD = 90 degrees
- Angle DOA = 90 degrees

4. **Using the Properties of the Circle**:
- In a circle, the angle subtended by a diameter at any point on the circle is a right angle (90 degrees).
- Therefore, we have:
- Angle ACB = 90 degrees
- Angle BDA = 90 degrees
- Angle DAC = 90 degrees
- Angle CBD = 90 degrees

5. **Proving the Sides are Equal**:
- Since A and C are endpoints of one diameter and B and D are endpoints of the other diameter, and both diameters are equal (as they are diameters of the same circle), we have:
- AC = BD (both are diameters)
- AB = AD = BC = CD (since they subtend equal angles at the center)

6. **Conclusion**:
- We have established that all sides of quadrilateral ABCD are equal (AB = BC = CD = DA) and all angles are right angles (90 degrees).
- Therefore, the quadrilateral ABCD is a square.

### Final Answer:
The quadrilateral formed by joining the endpoints of the two diameters that intersect at right angles is a **square**.