Introduction to Quadrilaterals Formed by Joining Midpoints of Squares

The question of which quadrilateral is formed when the midpoints of the corresponding sides of a square are joined has attracted interest from mathematicians and geometers for centuries. This article explores the fascinating properties that emerge from this intriguing geometry problem, providing a comprehensive understanding of the resultant quadrilateral, specifically focusing on when it is a square, rhombus, or parallelogram.

Formation and Properties of Quadrilaterals from Square Midpoints

When the midpoints of the sides of any quadrilateral are joined, the resulting shape is always a parallelogram. However, when dealing with a square, a specific transformation occurs. Each side of the original square is divided into two equal halves, and the midpoints are connected. This connection forms a new shape with several interesting properties.

Resulting Quadrilateral When Starting with a Square

When the midpoints of a square are joined, the result is a rhombus. To understand why, consider a square with each side of length 2x units. The diagonals of the square are the longest distances and are equal, given by 2√2x. When the midpoints are connected, each side of the resulting quadrilateral can be calculated using Pythagoras' theorem. Since the diagonals of the square are not necessarily equal to the sides of the resulting quadrilateral, it cannot be a square. Therefore, it must be a rhombus.

Detailed Explanation Using Pythagoras Theorem

Let's delve deeper into the mathematical reasoning behind this transformation. Each side of the original square is 2x units, so the distance between the midpoints of two adjacent sides can be calculated as follows:

Using the Pythagorean theorem, each side of the new quadrilateral is:

[asy] unitsize(1 cm); pair A, B, C, D, M, N, P, Q; A (0,0); B (2,0); C (2,2); D (0,2); M (A B)/2; N (B C)/2; P (C D)/2; Q (D A)/2; draw(A--B--C--D--cycle); draw(M--N--P--Q--cycle); label("2x", (A B)/2, S); label("2x", (B C)/2, E); label("2x", (C D)/2, N); label("2x", (D A)/2, W); label("M", (A B)/2, S); label("N", (B C)/2, E); label("P", (C D)/2, N); label("Q", (D A)/2, W); [/asy]

Each segment connecting the midpoints is actually the hypotenuse of a right triangle with legs of length x. Therefore, each new side has a length of:

[asy] unitsize(1 cm); pair A, B, C, D, M, N, P, Q; A (0,0); B (2,0); C (2,2); D (0,2); M (A B)/2; N (B C)/2; P (C D)/2; Q (D A)/2; draw(A--B--C--D--cycle); draw(M--N--P--Q--cycle); draw((A B)/2--(B D)/2, dashed); draw((B C)/2--(C A)/2, dashed); label("2x", (A B)/2, S); label("2x", (B C)/2, E); label("2x", (C D)/2, N); label("2x", (D A)/2, W); label("M", (A B)/2, S); label("N", (B C)/2, E); label("P", (C D)/2, N); label("Q", (D A)/2, W); label("x", (A B)/2--(B D)/2, W); label("x", (B C)/2--(C A)/2, W); [/asy]

Using the Pythagorean theorem:

[asy] unitsize(1 cm); pair A, B, C, D, M, N, P, Q; A (0,0); B (2,0); C (2,2); D (0,2); M (A B)/2; N (B C)/2; P (C D)/2; Q (D A)/2; label("2x", (A B)/2, S); label("2x", (B C)/2, E); label("2x", (C D)/2, N); label("2x", (D A)/2, W); label("M", (A B)/2, S); label("N", (B C)/2, E); label("P", (C D)/2, N); label("Q", (D A)/2, W); label("x", (A B)/2--(B D)/2, W); label("x", (B C)/2--(C A)/2, W); [/asy]

Since the sides are all equal, the new quadrilateral is a rhombus.

Area and Diagonal Lengths

Let's consider the specific case where each side of the original square is 2x units. The area of the original square is:

Area ( (2x)^2 4x^2 ) square units.

When the midpoints are joined, a smaller square is formed inside the original square. The side length of this smaller square is given by:

Side length ( sqrt{2} cdot x )

The area of the smaller square is:

Area ( (sqrt{2} cdot x)^2 2x^2 ) square units, which is half the area of the original square.

The diagonal of the smaller square is the same as the side length of the original square, which is 2x units, confirming the transformation properties.

Conclusion and Further Exploration

By joining the midpoints of a square, a rhombus is formed. This transformation provides a deeper insight into the geometric properties of squares and rhombi, as well as the relationships between different quadrilaterals. Further explorations can lead to a better understanding of how these shapes can be interconverted and the implications in various fields of mathematics and geometry.