a trapiziod has vertices A(-2,3) B(4,1) C(-1,-1) and D(2,-2). prove that the line joining the midpoints of the non-parallel sides is the average length of the two parallel sides of the trapaziod.
To prove that the line joining the midpoints of the non-parallel sides of the trapezoid is the average length of the two parallel sides, we can follow these steps:
1. Find the equations of the two parallel sides of the trapezoid.
2. Calculate the length of each parallel side.
3. Find the midpoints of the non-parallel sides.
4. Calculate the distance between the midpoints.
5. Compare the distance between the midpoints with the average length of the parallel sides.
Let's work through each step.
Step 1: Find the equations of the two parallel sides of the trapezoid.
The given vertices of the trapezoid are A(-2,3), B(4,1), C(-1,-1), and D(2,-2). The line AB is parallel to the line CD. We need to find the equations of both lines.
Using the point-slope form of a line, the equation of line AB is:
(y - y1) = m(x - x1)
(y - 3) = ((1-3)/(4-(-2))) * (x - (-2))
(y - 3) = -1/2 * (x + 2)
2y - 6 = -x - 2
x + 2y = 4
Similarly, the equation of line CD can be found using the coordinates C and D:
(y - y1) = m(x - x1)
(y - (-1)) = ((-2-(-1))/(2-(-1))) * (x - (-1))
(y + 1) = -1/3 * (x + 1)
3y + 3 = -x - 1
x + 3y = -4
Step 2: Calculate the length of each parallel side.
To find the length of the parallel sides AB and CD, we can use the distance formula.
For AB:
LengthAB = sqrt((4-(-2))^2 + (1-3)^2)
LengthAB = sqrt(6^2 + (-2)^2)
LengthAB = sqrt(36 + 4)
LengthAB = sqrt(40) = 2sqrt(10)
For CD:
LengthCD = sqrt((2-(-1))^2 + (-2-(-1))^2)
LengthCD = sqrt(3^2 + (-1)^2)
LengthCD = sqrt(9 + 1)
LengthCD = sqrt(10)
Step 3: Find the midpoints of the non-parallel sides.
To find the midpoint of BC, we take the average of the x-coordinates and the average of the y-coordinates:
MidpointBC = ((-1+2)/2, (-1-2)/2)
MidpointBC = (1/2, -3/2)
To find the midpoint of AD:
MidpointAD = ((-2+2)/2, (3+(-2))/2)
MidpointAD = (0, 1/2)
Step 4: Calculate the distance between the midpoints.
DistanceMidpoints = sqrt((1/2 - 0)^2 + (-3/2 - 1/2)^2)
DistanceMidpoints = sqrt((1/2)^2 + (-2)^2)
DistanceMidpoints = sqrt(1/4 + 4)
DistanceMidpoints = sqrt(17/4) = sqrt(17)/2
Step 5: Compare the distance between the midpoints with the average length of the parallel sides.
The average length of the parallel sides is (LengthAB + LengthCD) / 2.
AverageLengthParallelSides = (2sqrt(10) + sqrt(10)) / 2
AverageLengthParallelSides = sqrt(10)
Since DistanceMidpoints = sqrt(17)/2 and AverageLengthParallelSides = sqrt(10), we can see that they are not equal.
Therefore, based on the calculations, we cannot prove that the line joining the midpoints of the non-parallel sides is the average length of the two parallel sides of the trapezoid.