# Trapezoid and line connecting two sides

Base sides of trapezoid are 8 and 6. Parallel line to base sides that connects two other sides splits trapezoid to two surfaces with equal area. What is the length of this line?

My attempt: I tried drawing figure and manipulating formulas, to no avail for finding the length of this side.

Write area of each of the small trapezes in terms of the area of the original one. We use the bases of the original trapeze as $$L_{1,2}$$, the middle line is $$L$$, and the distance between $$L$$ and $$L_{1,2}$$ is $$h_{1,2}$$:$$\frac{L+L_1}{2}h_1=\frac12\frac{L_1+L_2}{2}(h_1+h_2)\\\frac{L+L_2}{2}h_2=\frac12\frac{L_1+L_2}{2}(h_1+h_2)$$ Now let's group terms with $$h_1$$ and $$h_2$$ on opposite sides of each equation: \begin{align}\left(\frac{L+L_1}{2}-\frac{L_1+L_2}{4}\right)h_1&=\frac{L_1+L_2}{4}h_2\\\frac{L_1+L_2}{4}h_1&=\left(\frac{L+L_2}{2}-\frac{L_1+L_2}{4}\right)h_2 \end{align} Simplifying the expressions in parenthesis, and taking the ratio of the two equations yields: $$\frac{2L+L_1-L_2}{L_1+L_2}=\frac{L_1+L_2}{2L+L_2-L_1}$$ This is equivalent to $$(2L+(L_2-L_1))(2L-(L_2-L_1))=(L_1+L_2)^2$$ or: $$4L^2-(L_2-L_1)^2=(L_1+L_2)^2$$ Then the final answer is $$L=\frac12\sqrt{(L_1+L_2)^2+(L_2-L_1)^2}=\sqrt{50}$$