Answer:
rhombus
Step-by-step explanation:
First we determine if the sides are parallel and if the angles are 90°. To do this, we find the slope of each side using the formula
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
For WX, we have
m = (4-1)/(-1-1) = 3/-2
For XY, we have
m = (1--2)/(1--1) = 3/2
For YZ, we have
m = (-2-1)/(-1--3) = -3/2
For ZW, we have
m = (1-4)/(-3--1) = -3/-2 = 3/2
Parallel lines have the same slope; this means that WX and YZ are parallel, as are XY and ZW.
Lines that make a 90° angle are called perpendicular lines, and they have slopes that are negative reciprocals of one another (meaning they have opposite signs and are flipped). None of these are negative reciprocals; this means none of the angles are 90°. This tells us this is not a square nor a rectangle.
To determine if this is a rhombus, we use the distance formula to find the length of each side:
[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]
The length of WX is
[tex]d=\sqrt{(4-1)^2+(-1-1)^2}=\sqrt{3^2+(-2)^2}=\sqrt{9+4}=\sqrt{13}[/tex]
The length of XY is
[tex]d=\sqrt{(1--2)^2+(1--1)^2}=\sqrt{(1+2)^2+(1+1)^2}=\sqrt{3^2+12^2}\\=\sqrt{9+4}=\sqrt{13}[/tex]
Since both pairs of opposite sides are parallel, this means opposite sides will be congruent; this tells us all four sides will be congruent, and this is a rhombus.
