Joining Midpoints in Parallelograms: A Geometrical Analysis
When you join the midpoints of the sides of a parallelogram, a new quadrilateral is formed known as the midpoint quadrilateral. This article explores the properties of this newly formed shape and why it is always a parallelogram, not necessarily a rhombus.
Definition and Formation of the Midpoint Quadrilateral
Consider a parallelogram with vertices denoted as A, B, C, and D. The midpoints of the sides AB, BC, CD, and DA are denoted as M, N, P, and Q, respectively. By connecting these midpoints, a new quadrilateral is created.
Vector Representation and Parallel Sides
In vector terms, if A and C are opposite vertices and B and D are the other pair of opposite vertices, the midpoints can be expressed as follows:
M (A B)/2 N (B C)/2 P (C D)/2 Q (D A)/2The sides of the new quadrilateral formed by these midpoints have vectors that demonstrate parallelism:
The vector MN (N - M) (B C)/2 - (A B)/2 (C - A)/2 is parallel to the vector PQ (P - Q) (D A)/2 - (C D)/2 (A - C)/2. The vector NP (P - N) (C D)/2 - (B C)/2 (D - B)/2 is parallel to the vector MQ (Q - M) D A)/2 - (A B)/2 (D - B)/2.Since the opposite sides are parallel, the figure formed is a parallelogram.
Conclusion
The quadrilateral formed by joining the midpoints of a parallelogram is always a parallelogram, not necessarily a rhombus unless the original parallelogram is a rhombus. In the case where the long side of the original parallelogram is twice that of the short side, you would get two rhombuses. However, joining the midpoints of any pair of sides of a rhombus will always result in two parallelograms.
Furthermore, joining the midpoints of a parallelogram creates a smaller parallelogram whose area is exactly half of the original one. This is due to the fact that each side of the new parallelogram is half the length of the corresponding side of the original one.