9515 1404 393
Step-by-step explanation:
Without loss of generality, we can let the intersection of the diagonals be the origin of our coordinate system, and the diagonals align with the x- and y-axes. We can define the half-diagonals of rhombus ABCD as 'a' and 'b', representing the distances from the origin to vertices A and B, respectively.
With A on the -x axis and vertices named clockwise, the slope of side AB will be b/a. When the rhombus is rotated 90° CCW, the slope of the side B'C' becomes a/b.
The midpoint of side AB is located at (-a/2, b/2), so the slope of the line from there through the origin is (b/2)/(-a/2) = -b/a.
So far, we have the slope to the midpoint of one side of the original rhombus is -b/a, and the slope of the side of the rotated rhombus is a/b. The product of these slopes is (-b/a)(a/b) = -1, so the two segments are perpendicular. So, we have shown a line through the center of a rhombus and the midpoint of one side is perpendicular to the side of the rhombus after it is rotated 90° about its center.