What is the length of the midsegment of the trapezoid made by the vertices A(0, 5), B(3, 3), C(5, -2) and D(-1, 2). Show equations and all work that leads to your answer.
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Checking the slopes of the lines, we see that the parallel bases are AB and CD. So, the sides are AD and BC.
The midpoint M of AD is (-1/2,7/2)
The midpoint N of BC is (4,1/2)
Now you have two points, so you can figure the length of the line MN
To find the length of the midsegment of the trapezoid, we first need to determine the midpoints of the two parallel sides of the trapezoid.
The midpoints of the two parallel sides can be found using the midpoint formula, which states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the coordinates ((x1 + x2)/2, (y1 + y2)/2).
Let's label the coordinates of the vertices:
A(0, 5), B(3, 3), C(5, -2), and D(-1, 2).
Now, let's find the midpoints of the two parallel sides:
The midpoint of AB is ((0 + 3)/2, (5 + 3)/2) = (3/2, 4).
The midpoint of CD is ((5 + (-1))/2, (-2 + 2)/2) = (2, 0).
Next, we need to find the distance between the midpoints of the two parallel sides.
The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula, which states that the distance d is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
Let's label the coordinates of the midpoints:
M1(3/2, 4) and M2(2, 0).
Using the distance formula, we can calculate the length of the midsegment:
d = sqrt((2 - 3/2)^2 + (0 - 4)^2)
= sqrt((-1/2)^2 + (-4)^2)
= sqrt(1/4 + 16)
= sqrt(17/4)
= sqrt(17)/2.
Therefore, the length of the midsegment of the trapezoid is sqrt(17)/2.