Plot the given points on a coordinate plane and graph the quadrilateral. Find the lengths of sides and the slopes to determine what kind of a quadrilateral that is.
We can see that the quadrilateral looks like a rhombus or a square. Recall that a rhombus is a parallelogram with all four sides congruent. A quadrilateral is a parallelogram if both pairs of opposite sides are parallel. Let's check if this is the case by finding the slopes using the Slope Formula.
Side
Points
y_2-y_1/x_2-x_1
Simplify
XW
X( 0,3), W( 3,0)
0- 3/3- 0
-1
WZ
W( 3,0), Z( 0, -3)
-3- 0/0- 3
1
ZY
Z( 0, -3), Y( -3, 0)
0-( -3)/-3- 0
-1
YX
Y( -3, 0), X( 0,3)
3- 0/0-( -3)
1
We can tell that the slopes of the opposite sides of our quadrilateral are equal. Therefore, both pairs of opposite sides are parallel and the quadrilateral is a parallelogram. We can also tell that the consecutive sides are perpendicular, as their slopes are opposite reciprocals.
1 ( - 1 ) = -1
Therefore, our quadrilateral is either a rectangle or a square. To check, we can find the lengths of its sides using the Distance Formula.
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Simplify
XW
X( 0,3), W( 3,0)
sqrt(( 3- 0)^2+( 0- 3)^2)
sqrt(18)
WZ
W( 3,0), Z( 0, -3)
sqrt(( 0- 3)^2+( -3- 0)^2)
sqrt(18)
ZY
Z( 0, -3), Y( -3, 0)
sqrt(( -3- 0)^2+( 0-( -3))^2)
sqrt(18)
YX
Y( -3, 0), X( 0,3)
sqrt(( 0-( -3))^2+( 3- 0)^2)
sqrt(18)
Our parallelogram has four congruent sides and these sides are perpendicular to each other. Therefore, it is a square. We want to find its area. The area of a square equals its side length squared, A= s^2. In our case s = sqrt(18).
