Answer: We can check if the lengths of opposite sides are equal. We see that DE and FG have the same length of 5, and DG and EF have the same length of sqrt(170)/sqrt(2), which can be simplified to 5*sqrt(2). Therefore, opposite sides are parallel and have equal length, and we can conclude that the quadrilateral DEFG is a parallelogram.
Step-by-step explanation: DE = sqrt(((-3)-(-8))^2 + ((6)-1)^2) = sqrt(25) = 5FG = sqrt((2-7)^2 + ((-1)-4)^2) = sqrt(50) = 5*sqrt(2)EF = sqrt((7-(-3))^2 + (4-6)^2) = sqrt(100) = 10DG = sqrt((2-(-8))^2 + ((-1)-1)^2) = sqrt(170)Next, we can use the slope formula to find the slopes of the sides:DE: slope = (6-1)/(-3-(-8)) = 5/5 = 1
FG: slope = ((-1)-4)/(2-7) = (-5)/(-5) = 1
EF: slope = (4-6)/(7-(-3)) = (-2)/10 = (-1)/5
DG: slope = ((-1)-1)/(2-(-8)) = (-2)/10 = (-1)/5
We can see that opposite sides FG and DE are parallel, as they have the same slope of 1, and opposite sides DG and EF are also parallel, as they have the same slope of -1/5.