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 should to check if it 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 DE: ( - 3,3), ( - 1,1)
1- 3/- 1-( - 3)
- 1
Slope of EF: ( - 1,1), (1,- 4)
- 4- 1/1-( - 1)
- 5/2
Slope of FG: (1,- 4), (- 3,0)
-(- 4)/- 3-1
- 1
Slope of GD: (- 3,0), ( - 3,3)
3-/- 3-(- 3)
Undefined
We got that the slopes of EF and GD are not equal, so these sides are not parallel. At the same time the slopes of DE and FG 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?
Let's recall that a trapezoid is isosceles if its non-parallel sides are congruent. Thus, we need to check whether the lengths of EF and GD are equal. To do this we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of EF: ( - 1,1), (1,- 4)
sqrt((1-( - 1))^2+(- 4- 1)^2)
sqrt(29)
Length of GD: (- 3,0), ( - 3,3)
sqrt(( - 3-(- 3))^2+( 3- )^2)
3
As we can see, the lengths of EF and GD are not equal, so these segments are not congruent. Therefore, DEFG is not an isosceles trapezoid.
