Final answer:
The length of the midsegment of the trapezoid with given vertices is found by calculating the midpoints of the non-parallel sides and then using the distance formula. The calculated length is 8 units.
Step-by-step explanation:
The vertices of the trapezoid are A(2,4), B(7,4), C(9,−1), and D(−2,−1). To find the length of the midsegment, we should identify the midpoints of the non-parallel sides AD and BC. The midsegment is the segment connecting these midpoints. The midpoint of AD can be found by averaging the x-coordinates and y-coordinates of A and D, resulting in the point (−0, 1.5). Similarly, the midpoint of BC is calculated to be the point (8, 1.5). The length of the midsegment is the distance between these two midpoints.
To calculate the distance between two points, we use the distance formula: distance = √((x2 − x1)^2 + (y2 − y1)^2). Substituting the midpoint coordinates into the formula, we get distance = √((8 − (−0))^2 + (1.5 − 1.5)^2) = √(8)^2 = 8. Thus, the length of the midsegment of the trapezoid is 8 units.