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?
$\endgroup$ 22 Answers
$\begingroup$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, ..
$\endgroup$ 0 $\begingroup$A parallelogram can be a
Rectangle if, each of the angle is a right angle, diagonals are equal.
Rhombus if, diagonals are perpendicular to each other.
Square if each angle is 90 degree, diagonals are perpendicular and equal.
