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  
\[x\]
.

We must find an equation in terms of  
\[x\]
  for the length  
\[l\]
  of the cutting line. When  
\[x=0, \: l=10\]
  and when  
\[x=4, \: l=4\]
.
From these two,  
\[l=10-1.5x\]

The area of the part above the cutting line is  
\[\frac{1}{2}(4+10-1.5x)(4-x)\]

The area of the part below the cutting line is  
\[\frac{1}{2}(10+10-1.5x)x\]

Equating these gives
\[\frac{1}{2}(4+10-1.5x)(4-x)=\frac{1}{2}(10+10-1.5x)x\]

Expanding this and simplifying gives  
\[3x^2-40x+56=0\]

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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