The coordinates of the vertices of quadrilateral defg are d(−2, 5) , e(2, 4) , f(0, 0) , and g(−4, 1) . which statement correctly describes whether quadrilateral defg is a rhombus? a. quadrilateral defg is not a rhombus because opposite sides are parallel but the four sides do not all have the same length. b. quadrilateral defg is not a rhombus because there are no pairs of parallel sides. c. quadrilateral defg is a rhombus because opposite sides are parallel and all four sides have the same length. d. quadrilateral defg is not a rhombus because there is only one pair of opposite sides that are parallel.

The correct option is a.Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length.

Explanation:

It is given that  coordinates of the vertices of quadrilateral DEFG are D(−2, 5) , E(2, 4) , F(0, 0) , and G(−4, 1) .

The distance formula for two points is given below,

Slope formula,

Slope of DE,

Slope of EF,

Slope of FG,

Slope of DG,

The slope of opposite sides are equal, it means the opposite sides are parallel.

All four sides do not have the same length. It means it is not a rhombus.

From the figure it is noticed that the opposite sides are parallel.

Therefore, the correct option is a.Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length.

a.Quadrilateral DEFG is not a rhombus because opposite sides are parallel but the four sides do not all have the same length.

Explanation:

To determine if the opposite sides are parallel, we find the slope of the line that passes through each pair of points.  The formula for slope is:

The slope of DE is:

m = (5-4)/(-2-2) = 1/-4 = -1/4

The slope of EF is:

m = (4-0)/(2-0) = 4/2 = 2

The slope of FG is:

m = (0-1)/(0--4) = -1/4

The slope of GD is:

m = (1-5)/(-4--2) = -4/-2 = 2

The slopes of opposite sides are parallel.

Now we must determine if the sides are congruent.  To do this, we use the distance formula:

The length of DE is:

The length of EF is:

The length of FG is:

The length of GD is:

Not all sides are congruent, so this is not a rhombus.