Erify that parallelogram abcd with vertices a (-5 -1),b(-9,6),c(-1,5) and d(3,-2) is a rhombus by showing it is a parallelogram with perpendicular diagonals

Slope of AC is (5-(-1))/(-1-(-5)) = 6/4 = 3/2
Midpoint of AC is ((-1-5)/2, (5-1)/2) = (-3, 2)

Slope of BD is (-2-6)/(3-(-9)) = -8/12 = -2/3
Midpoint of BD is ((3-9)/2, (-2+6)/2) = (-3, 2)

The diagonals bisect each other and their slopes are negative reciprocals. This is characteristic of a parallelogram that is a rhombus.