Answer
A parallelogram is a rhombus when the diagonals are perpendicular.
Work Step by Step
We assign coordinates to each point:
$A: 0,0 \\B: b,c \\ C: a+b, c \\ D: a,0$
We now find the slopes of each diagonal:
$m_1=\frac{c}{b-a}$
$m_2=\frac{c}{b+a}$
These must multiply to get -1, so:
$c^2 = a^2 - b^2 \\ c^2 +b^2 = a^2$
We now compare the lengths of each side using the distance formula:
$\sqrt{a^2} = \sqrt{b^2 + c^2}$
From the equation above, it follows that both sides of the equal sign are equal, so we see that when the diagonals are perpendicular, the parallelogram is a rhombus.
