Answer:
The quadrilateral ABCD is a rhombus.
Step-by-step explanation:
A rhombus is a quadrilateral having equal sides.
If ABCD is a rhombus then,
AB = BC = CD = DA
It is provided that the coordinates of the rhombus ABCD are:
A = (-1, -5)
B = (8, 2)
C = (11, 13)
D = (2, 6)
Use the distance formula to compute the lengths of AB, BC, CD and DA.
The distance formula is:
[tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/tex]
Compute the length of AB:
[tex]AB=\sqrt{(8-(-1))^{2}+(2-(-5))^{2}}=\sqrt{130}=11.4[/tex]
Compute the length of BC:
[tex]BC=\sqrt{(11-8)^{2}+(13-2)^{2}}=\sqrt{130}=11.4[/tex]
Compute the length of CD:
[tex]CD=\sqrt{(2-11)^{2}+(6-13)^{2}}=\sqrt{130}=11.4[/tex]
Compute the length of DA:
[tex]DA=\sqrt{(2-(-1))^{2}+(6-(-5))^{2}}=\sqrt{130}=11.4[/tex]
Thus, the lengths AB, BC, CD and DA are equal, i.e. all sides are of length 11.4.
Hence proved that the quadrilateral ABCD is a rhombus.
