4.1. The segment connecting the lateral sides of the trapezoid and parallel to its bases, which are 3 and 21, divides the trapezoid into two parts of equal area. Find the length of this segment.
Solution. 1st method. Let $a=3$ and $b=21$ be the lengths of the bases of the trapezoid. If $c$ is the length of the segment parallel to the bases, and $h_{1}$ and $h_{2}$ are the parts of the height of the trapezoid adjacent to the bases $a$ and $b$ respectively, then, by equating the areas, we get:

$$
\frac{a+c}{2} h_{1}=\frac{c+b}{2} h_{2}=\frac{1}{2} \cdot \frac{a+b}{2}\left(h_{1}+h_{2}\right) \Longleftrightarrow\left\{\begin{array}{l}
\frac{b+c}{a+c}=\frac{h_{1}}{h_{2}} \\
\frac{2(b+c)}{a+b}=1+\frac{h_{1}}{h_{2}}
\end{array}\right.
$$

From which

$$
\frac{2(b+c)}{a+b}=\frac{a+b+2 c}{a+c}
$$

Finally, we get $2 c^{2}=a^{2}+b^{2}$. Therefore

$$
c=\sqrt{\frac{a^{2}+b^{2}}{2}}=\sqrt{\frac{9+441}{2}}=15
$$

2nd method. Let $A B C D$ be a trapezoid with bases $A D=a=3, B C=b=21$ (see Fig. 1). Let $M N$ be the segment parallel to the bases. Let $F$ be the intersection point of the lines $A B$ and $C D$, the area of triangle $B C F$ is $a^{2} S$ and $M N=x$. Then

- from the similarity $M N F \sim B C F$ with the coefficient $k_{1}=x / a$ it follows: $S_{M N F}=k_{1}^{2} S_{B C F}=x^{2} S$;
- from the similarity $A D F \sim B C F$ with the coefficient $k_{2}=b / a$ it follows: $S_{A D F}=k_{2}^{2} S_{B C F}=b^{2} S$.

It remains to note that

$$
S_{A M N D}=S_{M B C N} \Longrightarrow\left(x^{2}-a^{2}\right) S=\left(b^{2}-x^{2}\right) S \Longrightarrow 2 x^{2}=a^{2}+b^{2}
$$

From which $x=\sqrt{\left(a^{2}+b^{2}\right) / 2}=15$.

Answer: 15. (C)

![](https://cdn.mathpix.com/cropped/2024_05_06_a8c31eb257d724803ce6g-05.jpg?height=394&width=557&top_left_y=223&top_left_x=498)

Fig. 1:

![](https://cdn.mathpix.com/cropped/2024_05_06_a8c31eb257d724803ce6g-05.jpg?height=394&width=557&top_left_y=223&top_left_x=1115)

Fig. 2:

![](https://cdn.mathpix.com/cropped/2024_05_06_a8c31eb257d724803ce6g-05.jpg?height=64&width=793&top_left_y=813&top_left_x=680)