The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle if the diagonals of PQRS are perpendicular.

Thus, the correct option is:

(c) diagonals of PQRS are perpendicular

Explanation:

• When you join the midpoints of the sides of any quadrilateral, the shape you get is called the Varignon parallelogram.
• For this parallelogram to be a rectangle, its angles must be right angles. This happens when the diagonals of the original quadrilateral are perpendicular, as it forces the angles to become 90°.

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Follow-up Questions:

1. What is the Varignon parallelogram, and how is it formed?
2. How can you prove that the quadrilateral formed is a rectangle when the diagonals are perpendicular?
3. What properties do diagonals of a quadrilateral need to satisfy for special cases (e.g., square, rhombus)?
4. Can a quadrilateral with unequal diagonals still form a rectangle by joining midpoints? Why or why not?
5. What happens if the diagonals of PQRS are both equal and perpendicular?

Tip: The properties of midpoints in quadrilaterals provide a geometric shortcut for understanding symmetry and shape transformations.