Answer:
Rhombus
Step-by-step explanation:
The given points are A(−5, 6), B(−1, 8), C(3, 6), D(−1, 4).
We use the distance formula to find the length of AB.
[tex]|AB|=\sqrt{(-1--5)^2+(8-6)^2}[/tex]
[tex]|AB|=\sqrt{16+4}[/tex]
[tex]|AB|=\sqrt{20}[/tex]
The length of AD is
[tex]|AD|=\sqrt{(-1--5)^2+(6-4)^2}[/tex]
[tex]|AD|=\sqrt{16+4}[/tex]
[tex]|AD|=\sqrt{20}[/tex]
The length of BC is:
[tex]|BC|=\sqrt{(-1-3)^2+(8-6)^2}[/tex]
[tex]|BC|=\sqrt{16+4}[/tex]
[tex]|BC|=\sqrt{20}[/tex]
The length of CD is
[tex]|CD|=\sqrt{(-1-3)^2+(6-4)^2}[/tex]
[tex]|CD|=\sqrt{16+4}[/tex]
[tex]|CD|=\sqrt{20}[/tex]
Since all sides are congruent the quadrilateral could be a rhombus or a square.
Slope of AB[tex]=\frac{8-6}{-1--5}=\frac{1}{2}[/tex]
Slope of BC [tex]=\frac{8-6}{-1-3}=-\frac{1}{2}[/tex]
Since the slopes of the adjacent sides are not negative reciprocals of each other, the quadrilateral cannot be a square. It is a rhombus
