Question

Consider the graph of quadrilateral WXYZ.

What is the most specific name for quadrilateral WXYZ?

square
rectangle
parallelogram
rhombus

Answer 1

rhombus

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

`m=(y_2-y_1)/(x_2-x_1)`

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:

`d=√((y_2-y_1)^2+(x_2-x_1)^2)`

The length of WX is

`d=√((4-1)^2+(-1-1)^2)=√(3^2+(-2)^2)=√(9+4)=√(13)`

The length of XY is

`d=√((1--2)^2+(1--1)^2)=√((1+2)^2+(1+1)^2)=√(3^2+12^2)\\=√(9+4)=√(13)`

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.