Final answer:
The length of the midsegment of the trapezoid ABCD is calculated using the midpoints of the trapezoid's legs and the distance formula; it is √26 units.
Step-by-step explanation:
The question asks for the length of the midsegment of a trapezoid given its vertices. The midsegment of a trapezoid is a line segment that connects the midpoints of the non-parallel sides (the legs) of the trapezoid and is parallel to the bases. To find this, we will identify the midpoints of the legs CD and AB of trapezoid ABCD and then use the distance formula to calculate the midsegment's length.
First, find the midpoint M1 of leg CD using the midpoint formula:
( (x1 + x2)/2, (y1 + y2)/2 )
Midpoint M1 = ( (9 - 2)/2, (-1 - 1)/2 ) = (7/2, -2/2) = (3.5, -1)
Similarly, find the midpoint M2 of leg AB.
Midpoint M2 = ( (7 + 2)/2, (4 + 4)/2 ) = (9/2, 8/2) = (4.5, 4)
Now, we use the distance formula to find the length of the midsegment M1M2:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Length of midsegment = √[(4.5 - 3.5)^2 + (4 - (-1))^2] = √[(1)^2 + (5)^2] = √[1 + 25] = √[26]
Therefore, the length of the midsegment of the trapezoid ABCD is √26 units.