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The image describes a problem that asks to prove that if two diameters of a circle are perpendicular to each other, the lines joining their endpoints form a square.

### Understanding the Problem:

1. **Given:** 
   - Two diameters of a circle are perpendicular to each other.
   - The endpoints of these diameters are connected, forming a quadrilateral.
   
2. **To Prove:** 
   - The quadrilateral formed by joining the endpoints of these perpendicular diameters is a square.

### Approach to the Proof:

1. **Step 1: Circle and Diameters**
   - Let the circle have a center at \( O \), and the radius of the circle be \( r \).
   - Assume the two diameters intersect at the center \( O \), and their endpoints are \( P, Q, R, S \).
   - The diameters are perpendicular to each other, meaning that \(\overline{PQ}\) is perpendicular to \(\overline{RS}\).

2. **Step 2: Symmetry and Geometry**
   - Since \( P, Q, R, S \) lie on the circle, the distances \( OP = OQ = OR = OS = r \) (the radii of the circle) are all equal.
   - As the diameters are perpendicular, each angle between the lines \( PQ \), \( QR \), \( RS \), and \( SP \) is 90 degrees.

3. **Step 3: Proving the Quadrilateral is a Square**
   - All sides of the quadrilateral \( PQRS \) are equal in length because the points are equally spaced along the circumference due to the symmetry of the diameters.
   - Since the quadrilateral has equal sides and angles of 90 degrees, it must be a square.

Thus, by the geometric properties of the circle and perpendicular diameters, the quadrilateral formed by joining the endpoints of the diameters is a square.

Would you like a more detailed explanation or additional questions on this topic?

### Related Questions:
1. What is the geometric significance of diameters in a circle?
2. How do perpendicular diameters divide a circle?
3. Can this property of perpendicular diameters forming a square be generalized for other polygons?
4. How does symmetry in a circle help in proving geometric theorems?
5. What are other properties of quadrilaterals formed inside a circle?

### Tip:
Whenever you deal with circle theorems, symmetry is a powerful tool that simplifies proofs and constructions.

In a circle, if any pair of diameters are perpendicular to each other, prove that the lines joining their ends form a square.

Prove that in a circle, if two diameters are perpendicular, the lines joining their endpoints form a square. This geometry problem involves key concepts like circle symmetry, quadrilateral properties, and the Pythagorean Theorem. Suitable for students in Grades 9-11.

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