Let's place F at the origin. Since AF and CD are vertical, the x-coordinate of A is 0, and the x-coordinates of C and D are the same. Let's use the Midpoint Formula to express the coordinates of B and E in terms of the coordinates of the vertices.
The midpoint between points (x_1,y_1) and (x_2,y_2) is ( x_1+x_22, y_1+y_22).
Let's substitute the coordinates of the vertices in this formula.
Points
Substitution
Midpoint
A(0,a) and C(p,c)
(0+p/2,a+c/2)
B(p/2,a+c/2)
F(0,0) and D(p,d)
(0+p/2,0+d/2)
E(p/2,d/2)
Since the first coordinates of B and E are the same, they lie on a vertical line. This proves that the midsegment BE is parallel to the bases.
To compare the length of the bases and the midsegment, we can use the fact that these segments are vertical, so the length is the difference between the y-coordinates of the endpoints.
Points
Distance
A(0,a) and F(0,0)
AF=a-0=a
B(p/2,a+c/2) and E(p/2,d/2)
BE=a+c/2-d/2=a+c-d/2
C(p,c) and D(p,d)
CD=c-d
We are asked to check the following equality.
BE=1/2(AF+CD)
Let's substitute the expressions from the table.
