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.

what are the bases of a trapezoid if the height is 15 meters and the midsegment is 32 meters

To find the length of the midsegment of the trapezoid, we first need to determine the midpoints of the bases.

1. Find the midpoint of AB:
- The midpoint formula is given by: ( (x1 + x2) / 2 , (y1 + y2) / 2 )
- Substitute the coordinates of A and B:
- Midpoint of AB = ( (0 + 3) / 2 , (5 + 3) / 2 )
= (3/2, 8/2)
= (3/2, 4)

2. Find the midpoint of CD:
- Substitute the coordinates of C and D into the midpoint formula:
- Midpoint of CD = ( (5 + (-1)) / 2 , (-2 + 2) / 2 )
= (4/2, 0)
= (2, 0)

Now that we have the midpoints of the bases, we can find the length of the midsegment by finding the distance between the two midpoints.

3. Use the distance formula:
- The distance formula is given by: sqrt( (x2 - x1)^2 + (y2 - y1)^2 )
- Substitute the coordinates of the midpoints:
- Distance = sqrt( (2 - 3/2)^2 + (0 - 4)^2 )
= sqrt( (4/4 - 3/2)^2 + (-4)^2 )
= sqrt( (-1/2)^2 + 16 )
= sqrt( (1/4) + 16 )
= sqrt( (1 + 64) / 4 )
= sqrt( 65 / 4 )
= sqrt( 65 ) / 2

Therefore, the length of the midsegment of the trapezoid formed by the given vertices is sqrt(65) / 2.