In the given figure we can determine the coordinate of point M from the graph, we get:
![M=(\frac{d}{2},\frac{c}{2})](https://tex.z-dn.net/?f=M%3D%28%5Cfrac%7Bd%7D%7B2%7D%2C%5Cfrac%7Bc%7D%7B2%7D%29)
We can also determine the coordinates of point N as:
![N=(\frac{a+b}{2},\frac{c}{2})](https://tex.z-dn.net/?f=N%3D%28%5Cfrac%7Ba%2Bb%7D%7B2%7D%2C%5Cfrac%7Bc%7D%7B2%7D%29)
Now, to determine the length of segment MN, we need to subtract the x-coordinate of M from the coordinates of N, we get:
![MN=\frac{a+b}{2}-\frac{d}{2}](https://tex.z-dn.net/?f=MN%3D%5Cfrac%7Ba%2Bb%7D%7B2%7D-%5Cfrac%7Bd%7D%7B2%7D)
Subtracting the fractions we get:
![MN=\frac{a+b-d}{2}](https://tex.z-dn.net/?f=MN%3D%5Cfrac%7Ba%2Bb-d%7D%7B2%7D)
Now, to obtain the length of AB we need to subtract the x-coordinate of A from the x-coordinate of B.
The coordinates of A are determined from the graph:
![A=(0,0)](https://tex.z-dn.net/?f=A%3D%280%2C0%29)
The coordinates of B are:
![B=(a,0)](https://tex.z-dn.net/?f=B%3D%28a%2C0%29)
Therefore, the length of segment AB is:
![AB=a](https://tex.z-dn.net/?f=AB%3Da)
Now we do the same procedure to determine the segment of CD. The coordinates of C are:
![C=(b,c)](https://tex.z-dn.net/?f=C%3D%28b%2Cc%29)
The coordinates of D are:
![D=(d,c)](https://tex.z-dn.net/?f=D%3D%28d%2Cc%29)
Therefore, CD is:
![CD=b-d](https://tex.z-dn.net/?f=CD%3Db-d)
Now, we determine MN as half the sum of the bases. The bases are AB and CD, therefore:
![MN=\frac{1}{2}(a+b-d)](https://tex.z-dn.net/?f=MN%3D%5Cfrac%7B1%7D%7B2%7D%28a%2Bb-d%29)
Therefore, we have proven that the median of a trapezoid equals half the sum of its bases.