Trapzium Cut in Half By Line Parallel to the Parallel Sides

Suppose we have a trapezium with parallel sides 10 and 4, and height 4. The trapezium is to be cut in half by a line parallel to the parallel sides. Let the distance of this line from the longer parallel side be  

We must find an equation in terms of  
  for the length  
  of the cutting line. When  
\[x=0, \: l=10\]
  and when  
\[x=4, \: l=4\]
From these two,  

The area of the part above the cutting line is  

The area of the part below the cutting line is  

Equating these gives

Expanding this and simplifying gives  

Solving this equation gives  
\[x= \frac{40 \pm \sqrt{928}}{6}\]

Obviously x must be less than the height of the trapezium so  
\[x= \frac{40 - \sqrt{928}}{6}= x= \frac{20 - 2\sqrt{58}}{3}\]
  since the other possibility gives  
\[x \gt 4\]

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