Prove please!?

Question

Why is it that the median of a trapezoid is:
a) parallel to the bases?
b) equal to the average of the two bases?

Answer 1

Start from basic figures the rectangle. The mid line of the rectangle is parellel to the bases and it is equal to 2* Length/2 = length if the mid line is parellel to is lengthy base. And if the mid line is parellel to breadth then it is equal to 2*Breadth/2= Breadth
Symbolically,the lengthof the mid line= 2*L/2=L or2*B/2=B
When the breadth line or length line of the rectangle is inclined we call it trapezoid. Please note here that even though sides are inclined position of mid line remains unaltered but the length of the line changes w.r.t. the change in inclination. I think this is enough to prove that the mid line of a trapezoid is always parellel to its bases.
Further, as the sides of the rectangle inclines to form a trapezium the it cut out two troangles on either sides of the rectangle. Also it cut out the mid lines. This cut out mid lines will be the mid lines of the triangles that are cut out. If a and b are the lengths of the cut out mid lines then the bases of such triangles that is parellel to its mid line is = 2a and 2b.
We know that length or breadth of a rectangle is L or B
Accordingly, Total length of the bases of the trapesium = (2L-2a-2b) or (2B-2a-2b)
=2(L-a-b) ; Here L-a-b = D which is the mid line of the trapezoid
ie = 2D
Threfore length of the mid line of the trapezoid is D which is half of 2D that is the sum of the bases of the trapezoid
Thus proved.
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Answer 2

The Parallel sides are called the bases. A median line is always parallel to the bases. The median line is halfway between the bases. The median divides the trapezoid into two smaller trapezoids each with half the altitude of the original.