3A. The trapezoid $ABCD$ is divided by a line $p$, such that $p\|AB\|CD$, into two parts of equal areas. Determine the length of the segment whose endpoints are the intersection points of the line $p$ and the non-parallel sides of the trapezoid, if the lengths of the bases $a$ and $b$ are given.
Solution. We continue the segments $A D$ and $B C$ until their intersection at $M$. Let $K L = p$. Then, $\triangle A B M \sim \triangle K L M \sim \triangle D C M$ and accordingly,

$$
\begin{aligned}
& P_{\triangle A B M}: P_{\triangle K L M}: P_{\triangle D C M} = a^{2}: m^{2}: b^{2}, \text{ i.e. } \\
& P_{\triangle A B M} = k a^{2}, P_{\Delta K L M} = k m^{2}, P_{\triangle D C M} = k b^{2}
\end{aligned}
$$

where $k$ is a proportionality coefficient. From the condition of the problem, we have

$$
\begin{aligned}
& P_{A B L K} = P_{K L C D} \\
& P_{\triangle A B M} - P_{\triangle K L M} = P_{\triangle K L M} - P_{\triangle D C M} \\
& k a^{2} - k m^{2} = k m^{2} - k b^{2} \\
& m^{2} = \frac{1}{2}\left(a^{2} + b^{2}\right) \\
& m = \overline{K L} = \sqrt{\frac{1}{2}\left(a^{2} + b^{2}\right)} .
\end{aligned}
$$

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