Finding the midpoint of a Quadrilateral
Given any quadrilateral, construct a midpoint on each side. Connect each consecutive midpoint with a segment. What are the properties of the shape formed by joining the midpoints? Does the resulting shape depend on the type of quadrilateral (e.g. convex & concave)? How does the area of the new figure compare to the area of the original quadrilateral?
Suppose you want the new figure to be a rectangle. What quadrilateral would you start with? What quadrilateral would you start with so that the new figure to be a rhombus? What quadrilateral would you start with so that the new figure is a square?
Will the new figure ever be non-convex?
No matter if the quadrilateral is concave, convex or imbedded, the shaped formed is a parallelogram. The reason is because the diagonal stay parallel to the midpoint connectors.
The areas formed are in a ratio of1:2. See the GSP sketch above.
In make the three conclusions below, I drew a square and created the quadrilateral using circles. I then manipulated this quadrilateral until I drew some conclusions about the square.
In order to form a square, the diagonals must be perpendicular and congruent. I found this by looking at a kite. Knowing the definition of a kite tells me that it is not a parallelogram but it does have a pair of congruent sides. This way I could form triangles. Low and behold, the triangles were congruent.
In order to form a rectangle, I would start with a quadrilateral whose diagonals are perpendicular to each other.
In order to form a rhombus the quadrilaterals diagonals must be congruent.
In order to form a square, the diagonals must be perpendicular and congruent. Which makes sense since a square is both a rhombus and rectangle.