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?
In order to verify that our quadrilateral is a trapezoid, we have to check if it has exactly one pair of parallel sides.
From the diagram, we can see that XW and YZ are two vertical segments, so they are parallel.
What about YX and WZ? It does not seem as though they are parallel, but let's make sure by calculating their slopes using the Slope Formula.
Side
Slope Formula
Simplified
Slope of WZ: ( 1,4), ( - 3,3)
3- 4/- 3- 1
1/4
Slope of YX: ( - 3,9), ( 1,8)
8- 9/1-( - 3)
- 1/4
We can see that the slopes of WZ and YX are not equal, so these sides are not parallel. Therefore, WXYZ has exactly one pair of parallel sides, which implies that it is a trapezoid.
Is It an Isosceles Trapezoid?
A trapezoid is isosceles if its non-parallel sides are congruent. Thus, we need to check whether the lengths of WZ and YX are equal. To do this we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of WZ: ( 1,4), ( - 3,3)
sqrt(( - 3- 1)^2+( 3- 4)^2)
sqrt(17)
Length of YX: ( - 3,9), ( 1,8)
sqrt(( 1-( - 3))^2+( 8- 9)^2)
sqrt(17)
We got that WZ and YX are each sqrt(17) units long, which means that they are congruent. Therefore, WXYZ is an isosceles trapezoid.
