Given that PQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?
We can conclude that PQRS is a rectangle.
Reasoning: A quadrilateral whose diagonals bisect each other is a parallelogram. Since PQRS is also a cyclic quadrilateral, it must be a rectangle. This is because opposite angles of a cyclic quadrilateral sum to 180°, and opposite angles of a parallelogram are equal. Therefore, each angle equals 90°, making it a rectangle.
Explanation
This question tests deductive reasoning by combining two properties: (1) The property that diagonals bisecting each other defines a parallelogram, and (2) The property that a cyclic parallelogram must be a rectangle. The textbook context emphasizes how proofs are constructed through 'valid arguments' and 'previously proved results.' Students apply known geometric theorems to reach a logical conclusion. The key insight is that a parallelogram inscribed in a circle must have all angles equal to 90°, since opposite angles of a cyclic quadrilateral sum to 180° and opposite angles of a parallelogram are equal.
Solution Steps
Step 1: Identify that diagonals bisecting each other means PQRS is a parallelogram.
Step 2: Apply the cyclic quadrilateral property - opposite angles sum to 180°.
Step 3: Apply the parallelogram property - opposite angles are equal.
Step 4: Combine both properties: if opposite angles are equal and sum to 180°, each angle = 90°.
Step 5: Conclude that PQRS is a rectangle (parallelogram with all angles 90°).