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[ Parallel lines, properties and signs. Secants]

Points $M$ and $K$ are located on the side $A B$, and points $N$ and $L$ are on the side $C D$ of trapezoid $A B C D$, with $M N \| A D$. It is known that the areas of trapezoids $M B C N$ and $M A D N$ are in the ratio $1: 5$. Find $M N$ and $K L$, if $B C=$ $a, A D=b$.

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Using additional constructions, obtain similar triangles.

## Solution

See the solution to problem $\underline{116358}$.

## Answer

$\sqrt{\frac{5 a^{2}+b^{2}}{6}}$