Answer:
Yes, it is a rhombus because the slope of diagonal segment BD equals 2, the slope of diagonal segment AC equals -1/2, and the diagonals bisect each other.
Step-by-step explanation:
The slope of diagonal BD is ...
slope BD = (change in y)/(change in x) = (-4/-2) = 2
The slope of diagonal AC is ...
slope AC = (change in y)/(change in x) = (4/-8) = -1/2
The midpoint of BD is ...
((-4, 5) +(-6, 1))/2 = (-10, 6)/2 = (-5, 3)
The midpoint of AC is ...
((-1, 1) +(-9, 5))/2 = (-10, 6)/2 = (-5, 3)
so the diagonals bisect each other.
While the third statement is true (both midpoints are the same), that fact alone is not sufficient to allow the figure to be declared a rhombus. The diagonals must also be perpendicular (have slopes whose product is -1). The appropriate answer choice is the <em>second one</em>:
- Yes, it is a rhombus because the slope of diagonal segment BD equals 2, the slope of diagonal segment AC equals -1/2, and the diagonals bisect each other.
