Question

Statement (1): The diagonals of a quadrilateral are perpendicular to each other; P, Q, R and S are mid-points of sides AB, BC, CD and DA, respectively. Then PQRS will be a rectangle.

Statement (2): Quadrilateral PQRS will be a square as each of its angles will be 90°.

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Statement 1 is true, and statement 2 is false.

Explanation:

Statement 1 is true because if the diagonals of the outer quadrilateral are perpendicular, the sides of the inner midpoint quadrilateral PQRS meet at right angles, making it a rectangle. Statement 2 is false because PQRS is only a rectangle, not a square, unless the diagonals of the original quadrilateral are also equal in length.