To prove that the quadrilateral $PQRS$ is a rhombus, let's go through the following steps:

Step 1: Understanding the given figure and the properties

You are given a rectangle $ABCD$, and points $P, Q, R, S$ are the midpoints of the sides of the rectangle:

• $P$ is the midpoint of side $AB$
• $Q$ is the midpoint of side $BC$
• $R$ is the midpoint of side $CD$
• $S$ is the midpoint of side $DA$

The diagonals $AC$ and $BD$ of the rectangle intersect at point $O$, and we need to prove that quadrilateral $PQRS$ formed by joining midpoints of the sides is a rhombus.

Step 2: Properties of a rhombus

To prove that quadrilateral $PQRS$ is a rhombus, we must show that:

• All four sides of $PQRS$ are equal in length.
• The diagonals of $PQRS$ bisect each other at right angles (90°).

Step 3: Proving that all sides are equal (congruence of sides)

Midpoint Theorem:

Since $P, Q, R, S$ are midpoints of sides $AB, BC, CD, DA$ of rectangle $ABCD$, the segment joining two midpoints is parallel to the diagonal of the rectangle and half its length.

• $PQ \parallel AC$ and $PQ = \frac{1}{2}AC$
• $QR \parallel BD$ and $QR = \frac{1}{2}BD$
• $RS \parallel AC$ and $RS = \frac{1}{2}AC$
• $SP \parallel BD$ and $SP = \frac{1}{2}BD$

Since $AC$ and $BD$ are the diagonals of the rectangle, which are equal in length, we can conclude that $PQ = RS$ and $QR = SP$, implying that the opposite sides of quadrilateral $PQRS$ are equal in length.

Step 4: Proving that the diagonals bisect at 90° (Perpendicular diagonals)

The diagonals of rectangle $ABCD$ bisect each other at $O$ and intersect at right angles because they are perpendicular in the rectangle. Therefore, diagonals $AC$ and $BD$ are perpendicular, and since $PQ \parallel AC$ and $QR \parallel BD$, the diagonals of quadrilateral $PQRS$ (i.e., $PR$ and $QS$) will also be perpendicular.

Conclusion:

Since the opposite sides of quadrilateral $PQRS$ are equal and its diagonals are perpendicular, $PQRS$ is a rhombus.

Would you like more details or clarification on any of the steps? Here are some follow-up questions:

1. How does the Midpoint Theorem apply to other quadrilaterals?
2. Can this proof be generalized to other shapes like parallelograms?
3. What are the differences between a rhombus and other quadrilaterals like squares and rectangles?
4. How can you use coordinate geometry to prove that $PQRS$ is a rhombus?
5. What is the role of the diagonals in determining the properties of a rhombus?

Tip: Always ensure to label important points (like midpoints and intersection points) in geometry problems as it can simplify the proof process!