The given points are A(–4, –7), B(–1, 2), C(8, 5) and D(5, –4). 

Distance between two points = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)

According to the distance formula,

\[AB =\sqrt{([-1-(-4)]^2 +[2-(-7)]^2}\]

\[∴ AB = \sqrt{(9+81) }\]

\[∴ AB = \sqrt{90                      ...(1)}\]

\[BC =\sqrt{([8-(-1)]^2+(5-2)^2)}\]

\[∴ BC =\sqrt{(9^2+3^2)}\]

\[∴ BC = \sqrt{(81+9)}\]

\[∴ BC = \sqrt{90                     ...(2)}\]

\[CD = \sqrt{((5-8)^2 +(-4-5)^2)}\]

\[∴ CD =\sqrt{((-3)^2 +(-9)^2)}\]

\[ ∴ CD =\sqrt{(9+81)}\]

\[∴ CD = \sqrt{90           ....... (3)}\]

\[ AD = \sqrt{([5-(-4)]^2+ [-4-(-7)]^2)}\]

\[∴ AD =\sqrt{(9^2+3^2)}\]

\[∴ AD =\sqrt{(81+9)}\]

\[∴ AD = \sqrt{90                      ....... (4)}\]

From (1), (2), (3), and (4)}\]

AB = BC = CD = AD

Thus, all sides are equal.

In a quadrilateral, if all the sides are equal, then it is a rhombus.

∴ square ABCD is a rhombus.

∴ Points A, B, C and D are the vertices of rhombus ABCD.