First we will verify that it is a trapezoid, and then we will determine whether the figure is an isosceles trapezoid.
Is It a Trapezoid?
To verify that our quadrilateral is a trapezoid, we have to check if our quadrilateral has exactly one pair of parallel sides. To do this, let's find the slope of each side using the Slope Formula.
Side
Slope Formula
Simplified
Slope of WX: ( -5,-1), ( -2,2)
2-( -1)/-2-( -5)
1
Slope of XY: ( -2,2), ( 3, 1)
1- 2/3-( -2)
- 1/5
Slope of YZ: ( 3,1), ( 5, -3)
-3- 1/5- 3
- 2
Slope of ZW: ( 5, -3), ( -5,-1)
-1-( -3)/-5- 5
- 1/5
We can see that the slopes of WX and YZ are not equal, so these sides are not parallel. The slopes of XY and ZW are equal, so these sides are parallel. Since our quadrilateral has exactly one pair of parallel sides, it is a trapezoid.
Is It an Isosceles Trapezoid?
A trapezoid is isosceles if its non-parallel sides are congruent. Therefore, we want to check whether the lengths of WX and YZ are equal. To do this, we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of WX: ( -5,-1), ( -2,2)
sqrt(( -2-( -5))^2+( 2-( -1))^2)
sqrt(18)
Length of YZ: ( 3,1), ( 5, -3)
sqrt(( 5- 3)^2+( -3- 1)^2)
sqrt(20)
Since the lengths are not equal, WX and YZ are not congruent. Therefore, WZYZ is not an isosceles trapezoid.
