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 determine if our quadrilateral is a trapezoid, we have 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 AB: ( - 3,3), ( - 4,- 1)
- 1- 3/- 4-( - 3)
4
Slope of BC: ( - 4,- 1), ( 5,- 1)
- 1-( - 1)/5-( - 4)
0
Slope of CD: ( 5,- 1), ( 2,3)
3-( - 1)/2- 5
- 4/3
Slope of DA: ( 2,3), ( - 3,3)
3- 3/- 3- 2
0
We can see that the slopes of AB and CD are not equal, so these sides are not parallel. The slopes of BC and DA 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 AB and CD are equal. To do this, we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of AB: ( - 3,3), ( - 4,- 1)
sqrt(( - 4-( - 3))^2+( - 1- 3)^2)
sqrt(17)
Length of CD: ( 5,- 1), ( 2,3)
sqrt(( 2- 5)^2+( 3-( - 1))^2)
5
Since the lengths are not equal, AB and CD are not congruent. Therefore, ABCD is not an isosceles trapezoid.
