The question is a follows.
Midpoints of the sides of a quadrilateral are the vertices of a paralleogram. Determine under what conditions this parallelogram will be (1) a rectangle, (2) a rhombus, (3) a square.
I tried this for my hw question but I am not sure if it was right. For (1), I think the two diagonals have to be perpendicular to each other, for (2), I wrote that the quadrilateral has to be rectangle. And for (3) I wrote that the quadrilateral has to be both (1) and (2). Can someone explain if this argument is right?? And if possible, can anyone give simple proof for this?
Best Answer
1: The parallelogram is a rhombus if and only if the diagonals of the quadrilateral are perpendicular, that is, if the quadrilateral is an orthodiagonal quadrilateral
2: The parallelogram is a rectangle if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an equidiagonal quadrilateral. Reference for those two theorems.
3 (Hint): When the quadrilateral is both a rectangle and a rhombus, then it is a square. So, ..