Place the trapezoid in the coordinate plane so that one base is on the x-axis.
See solution.
Practice makes perfect
We are asked to prove that the median of an isosceles trapezoid is parallel to the bases.
We are asked to use a coordinate proof, so let's place the trapezoid in the coordinate plane so that one base is on the x-axis and one vertex is at the origin.
Since side BC is parallel to the x-axis, the y-coordinates of B and C are the same.
The median connects the midpoints of the legs, so let's recall the Midpoint Formula.
The midpoint
between points(x_1,y_1)and(x_2,y_2)is
(x_1+x_2/2,y_1+y_2/2).
Let's use this formula to express the coordinates of the midpoints of the legs in terms of the coordinates of the vertices.
Points
Substitution
Midpoint
A(0,0) and B(b,p)
(0+b/2,0+p/2)
E(b/2,p/2)
C(c,p) and D(d,0)
(c+d/2,p+0/2)
F(c+d/2,p/2)
Since the second coordinates of E and F are the same, the line connecting them is horizontal.
This means that the median of a trapezoid is indeed parallel to the bases.
Extra
We did not use that the trapezoid is isosceles.
Note that in the proof we did not use that the trapezoid is isosceles.
The midsegment in any trapezoid is parallel to the bases, not only in isosceles trapezoids.
