Prove algebraically that the line joining the midpoints of the nonparallel sides of a trapezoid is parallel to the bases of the trapezoid.
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To prove that the line joining the midpoints of the nonparallel sides of a trapezoid is parallel to the bases, we can use the concept of slope.
Let's consider a trapezoid ABCD, where AB and CD are the parallel bases, and E and F are the midpoints of the nonparallel sides AD and BC, respectively.
To start, let's find the slopes of the line segments AE and BF. The slope of a line can be found using the formula:
slope = (change in y-coordinates) / (change in x-coordinates).
Let's assume the coordinates of points A, B, C, D, E, and F are as follows:
A(x1, y1)
B(x2, y2)
C(x3, y3)
D(x4, y4)
E(x5, y5)
F(x6, y6)
Now, we can find the slopes of the line segments AE and BF.
The slope of AE, denoted as m1, can be calculated as:
m1 = (y5 - y1) / (x5 - x1)
The slope of BF, denoted as m2, can be calculated as:
m2 = (y6 - y2) / (x6 - x2)
If the line joining the midpoints of the nonparallel sides is parallel to the bases, then the slopes of AE and BF should be equal.
Therefore, we need to prove that m1 = m2.
Now, let's expand the equations for m1 and m2:
m1 = (y5 - y1) / (x5 - x1) ...(1)
m2 = (y6 - y2) / (x6 - x2) ...(2)
Now, let's simplify equation (1) further:
m1 = (y5 - y1) / (x5 - x1)
= (y5 - y1) / (x5 - x1) * (x2 - x2) / (x2 - x2)
= (y5 - y4) / (x5 - x4) ...(3)
Similarly, let's simplify equation (2):
m2 = (y6 - y2) / (x6 - x2)
= (y3 - y2) / (x3 - x2) ...(4)
Since AB and CD are parallel bases, their slopes are equal. Therefore, the slope of AB, denoted as m3, is:
m3 = (y2 - y1) / (x2 - x1)
Notice that both the slopes m1 (equation 3) and m3 are equal, as they represent the slopes of parallel lines. Similarly, both the slopes m2 (equation 4) and m3 are equal.
Hence, we can conclude that m1 = m2. Therefore, the line joining the midpoints of the nonparallel sides of a trapezoid is parallel to the bases of the trapezoid.