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 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: ( - 5,6), ( 4,9)
9- 6/4-( - 5)
1/3
Slope of BC: ( 4,9), (4,4)
4- 9/4- 4
Undefined
Slope of CD: (4,4), (- 2,2)
2-4/- 2-4
1/3
Slope of DA: (- 2,2), ( - 5,6)
6-2/- 5-(- 2)
- 4/3
We can see that the slopes of BC and DA are not equal, so these sides are not parallel. At the same time, the slopes of AB and CD are equal, which implies that these sides are parallel. Since ABCD 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 BC and DA are equal. To do this we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of BC: ( 4,9), (4,4)
sqrt((4- 4)^2+(4- 9)^2)
5
Length of DA: (- 2,2), ( - 5,6)
sqrt(( - 5-( - 2))^2+( 6- 2)^2)
5
We got that BC and DA are each 5 units long, which means that they are congruent. Therefore, ABCD is an isosceles trapezoid.
