In rhombus ABCD, with P, Q, R, and S as mid-points of sides AB, BC, CD, and DA respectively:
PQRS is a Parallelogram: By the Midpoint Theorem, PQ and SR are parallel and equal as they join midpoints of consecutive sides of the rhombus. Similarly, PS and QR are parallel and equal. Thus, PQRS is a parallelogram.
Angles in PQRS are Right Angles: In a rhombus, diagonals bisect each other at right angles. Since P, Q, R, and S are midpoints, they form right angles at the intersection of the diagonals.
Therefore, PQRS is a parallelogram with right angles, making it a rectangle.
Let’s discuss in detail
Rhombus ABCD and Midpoints
In the rhombus ABCD, points P, Q, R, and S are the midpoints of the sides AB, BC, CD, and DA, respectively. A rhombus is a special type of parallelogram where all sides are equal, and its diagonals bisect each other at right angles. The positioning of midpoints on the sides of a rhombus creates a unique geometric configuration. This setup allows for the exploration of the properties of the quadrilateral formed by connecting these midpoints, leading to insights into the nature of this quadrilateral, PQRS, within the context of the properties of a rhombus.
PQRS as a Parallelogram
Firstly, it’s important to establish that PQRS is a parallelogram. According to the Midpoint Theorem, a line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Applying this theorem to the triangles formed by the diagonals of the rhombus, it can be deduced that PQ is parallel to SR and PS is parallel to QR. Since opposite sides are parallel, PQRS satisfies one of the key criteria for being a parallelogram.
Equal Lengths of Opposite Sides in PQRS
In addition to being parallel, the opposite sides of PQRS are also equal in length. This is again a result of the Midpoint Theorem. In a rhombus, all sides are equal, so when midpoints are connected, the lengths of PQ and SR, and PS and QR, are equal because they each span half the length of the sides of the rhombus. This equality of opposite sides further confirms that PQRS is a parallelogram.
Right Angles in PQRS
A defining feature of a rectangle is the presence of right angles. In a rhombus, the diagonals bisect each other at right angles. Since P, Q, R, and S are midpoints of the sides of the rhombus, they lie on the diagonals. Therefore, at the point where the diagonals intersect, which is inside PQRS, right angles are formed. This means that each angle in PQRS is a right angle.
PQRS as a Rectangle
Given that PQRS is a parallelogram with right angles, it meets the definition of a rectangle. A rectangle is a parallelogram with right angles, and since PQRS has been shown to have both these properties, it can be conclusively stated that PQRS is a rectangle. This conclusion is a direct result of the properties of the rhombus ABCD and the specific positioning of the midpoints on its sides.
Geometric Nature of PQRS
In conclusion, the quadrilateral PQRS, formed by connecting the midpoints of the sides of rhombus ABCD, is a rectangle. This is established through the application of the Midpoint Theorem and the properties of a rhombus, specifically the behavior of its diagonals. The exercise demonstrates how specific configurations within a geometric figure can lead to the formation of another figure with its own distinct properties, showcasing the interconnectedness and elegance of geometric principles.