1.35. On the lateral sides $AB$ and $CD$ of trapezoid $ABCD$, points $M$ and $N$ are taken such that segment $MN$ is parallel to the bases and divides the area of the trapezoid in half. Find the length of $MN$, if $BC=a$ and $AD=b$.
1.35. Let $M N=x ; E$ be the intersection point of lines $A B$ and $C D$. Triangles $E B C, E M N$ and $E A D$ are similar, so $S_{E B C}: S_{E M N}: S_{E A D}=$ $=a^{2}: x^{2}: b^{2}$. Since $S_{E M N}-S_{E B C}=S_{M B C N}=$ $=S_{M A D N}=S_{E A D}-S_{E M N}$, then $x^{2}-a^{2}=b^{2}-x^{2}$, i.e. $x^{2}=\left(a^{2}+b^{2}\right) / 2$.