In order to verify that our quadrilateral is a trapezoid, we need to check if it has exactly one pair of parallel sides. Recall that if two segments are parallel their slopes are the same. Let's find the slope of each side using the Slope Formula.
Side
Slope Formula
Simplified
Slope of MN: ( - 2,0), ( 0,4)
4- 0/0-( - 2)
2
Slope of NP: ( 0,4), (5,4)
4- 4/5- 4
0
Slope of PQ: (5,4), (8,0)
-4/8-5
- 4/3
Slope of QM: (8,0), ( - 2,0)
0-/8-(- 2)
0
We can conclude that NP and QM are parallel, while MN and PQ are not. Since our quadrilateral has exactly one pair of parallel sides, it is a trapezoid.
Is It an Isosceles Trapezoid?
Let's review that a trapezoid is isosceles if its non-parallel sides are congruent. Hence, we need to check whether the lengths of MN and PQ are equal. To do this we will use the Distance Formula.
Side
Distance Formula
Simplified
Length of MN: ( - 2,0), ( 0,4)
sqrt(( 0-( - 2))^2+( 4- 0)^2)
sqrt(20)
Length of PQ: (5,4), (8,0)
sqrt((8-5)^2+( -4)^2)
5
We can see that the length of MN and PQ are not the same, so these segments are not congruent. Therefore, MNPQ is not an isosceles trapezoid.
