Explanation:
The quadrilateral will be a trapezoid if it has exactly one pair of parallel sides. It would be isosceles if the lengths of the non-parallel sides were the same. To make the required proof, we will show two sides parallel (have the same slope) and the other two sides of different length.
It is convenient to consider the differences between the coordinates at the ends of each line segment. Here, we can use FG to mean (G -F), for example. The slope of the segment will be the ratio of the y-difference to the x-difference. The length of the segment will be the root of the sum of the squares of the differences.
EF = (-2, 0) -(-3, -5) = (1, 5) . . . . slope = 5/1 = 5; length = √(1²+5²) = √26
FG = (2, 3) -(-2, 0) = (4, 3) . . . . slope = 3/4; length = √(4²+3²) = 5
GH = (5, 1) -(2, 3) = (3, -2) . . . . slope = -2/3; length = √(3² +2²) = √13
HE = (-3, -5) -(5, 1) = (-8, -6) . . . . slope = -6/-8 = 3/4; length = √(8² +6²) = 10
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We observe that segments FG and HE both have a slope of 3/4, so are parallel. No other segment slopes are the same, so there is exactly one pair of parallel segments.
Every segment is a different length from every other, so the trapezoid is not isosceles.
