Math Forum - Project of the Month, February 1997
A Math Forum Project
February POM - Winner
Sequoia Middle School, Pleasant Hill, California
Let the quadrilateral be ABCD.
Let E, F, G, H be the midpoints of sides AB, BC, CD, DA respectively.
AC and BD are the two diagonals.
Since E and F are midpoints of Triangle ABC, EF//AC.
Similarly, HG//AC because H and G are midpoints of Triangle DAC.
Using the same argument for Triangles ABD and CBD, we have EH//BD and FG//BD.
Therefore, for any quadrilateral, the quadrilateral formed by connecting
the consecutive midpoints must be at least a PARALLELOGRAM with the
length half the length of the corresponding diagonals.
- If the two diagonals AC and BD are of equal length, then the
parallelogram would be equilateral or a RHOMBUS.
- If the two diagonals are perpendicular to each other, then the
parallelogram would be a RECTANGLE.
- Of course, if the diagonals are both equal length and perpendicular
to each other, the parallelogram would be a SQUARE.
To answer your questions:
- Rectangle - the parallelogram is equilateral or a Rhombus (case 1)
- Kite - it is a rectangle (case 2)
- Isosceles Trapezoid - Rhombus (case 1)
- Square - Square (case 3)
- Rhombus - Rectangle (case 2)
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