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 a trapezoid is the average length of the two parallel sides, we can follow these steps:
1. Find the coordinates of the midpoints of the non-parallel sides:
- Let M be the midpoint of side AB (non-parallel).
- Let N be the midpoint of side CD (non-parallel).
To find the midpoint of a line segment, you can use the midpoint formula:
- The coordinates of the midpoint M are [(x1 + x2) / 2, (y1 + y2) / 2] where (x1, y1) and (x2, y2) are the coordinates of the endpoints.
2. Find the lengths of the two parallel sides:
- Let a be the length of side AB (parallel to CD).
- Let b be the length of side CD.
To find the length of a line segment, you can use the distance formula:
- The length of a line segment with endpoints (x1, y1) and (x2, y2) is √[(x2 - x1)^2 + (y2 - y1)^2].
3. Calculate the average length of the two parallel sides:
- Average length = (a + b) / 2.
4. Compare the length of the line segment MN with the average length of the two parallel sides:
- If MN = Average length, then we have proved the statement.
Let's apply these steps to the given trapezoid with vertices A(-2,3), B(4,1), C(-1,-1), and D(2,-2).
1. Find the coordinates of the midpoints:
- Coordinates of M = [(-2 + 4) / 2, (3 + 1) / 2] = [1, 2].
- Coordinates of N = [(-1 + 2) / 2, (-1 - 2) / 2] = [0.5, -1.5].
2. Find the lengths of the two parallel sides:
- Length of AB = √[(4 - (-2))^2 + (1 - 3)^2] = √(6^2 + (-2)^2) = √(36 + 4) = √40 ≈ 6.325.
- Length of CD = √[(2 - (-1))^2 + (-2 - (-1))^2] = √(3^2 + (-1)^2) = √(9 + 1) = √10 ≈ 3.162.
3. Calculate the average length of the two parallel sides:
- Average length = (6.325 + 3.162) / 2 = 9.487 / 2 ≈ 4.7435.
4. Compare the length of MN with the average length:
- Length of MN = √[(1 - 0.5)^2 + (2 - (-1.5))^2] = √(0.5^2 + 3.5^2) = √(0.25 + 12.25) = √12.5 ≈ 3.5355.
Since MN ≈ 3.5355 is approximately equal to the average length of 4.7435, we have proved that the line joining the midpoints of the non-parallel sides is the average length of the two parallel sides of the trapezoid.