Question

If A(−1, 1), B(−4, 5), C(−9, 5), and D(−6, 1) are the vertices of a quadrilateral, do the points form a rhombus? Justify your answer.

Yes, it is a rhombus because the slope of diagonal segment BD equals negative two and the slope of diagonal segment AC equals start fraction one over two end fraction full stop
Yes, it is a rhombus because the slope of diagonal segment BD equals two, the slope of diagonal segment AC equals negative start fraction one over two end fraction, and the diagonals bisect each other.
Yes, it is a rhombus because the midpoint of diagonals segment BD and segment AC is (−5, 3).
No, it is not a rhombus because the slope of diagonal segment BD equals two and the slope of diagonal segment AC equals negative start fraction one over two end fraction full stop

Answer 1

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.

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 second one:

• 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.