[ Similar figures ]  $[\underline{\text { Mean }}$ values $]$

A line parallel to the bases of a trapezoid divides it into two similar trapezoids.

Find the segment of this line contained within the trapezoid, if the bases are equal to $a$ and $b$.

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Let a line parallel to the bases $AD = a$ and $BC = b$ of trapezoid $ABCD$ intersect the lateral sides $AB$ and $CD$ at points $M$ and $N$ respectively, and trapezoids $MBCN$ and $AMND$ are similar. Then $BC : MN = MN : AD$, from which $MN^2 = AD \cdot BC = ab$. Therefore, $MN = \sqrt{ab}$.