Task 2. Two lines $p$ and $q$ are parallel to the bases of the trapezoid $ABCD$ and divide the leg $AD$ into three equal parts. Determine the lengths of the segments $x$ and $y$ of the lines $q$ and $p$ between the legs, if $\overline{AB}=13$ and $\overline{CD}=4$.
Solution. Let $p$ intersect the sides $A D$ and $B C$ at points $E$ and $F$, respectively, and let $q$ intersect the sides $A D$ and $B C$ at points $G$ and $H$, respectively (diagram on the right). Then, $EF$ is the midline of trapezoid $G H C D$, so

![](https://cdn.mathpix.com/cropped/2024_06_05_1b044dba0a21c223c4c1g-8.jpg?height=410&width=736&top_left_y=396&top_left_x=764)

$$
y=\overline{E F}=\frac{1}{2}(\overline{G H}+\overline{C D})=\frac{1}{2}(x+4)
$$

From $G H$ being the midline of trapezoid $A B F E$, we get

$$
x=\overline{G H}=\frac{1}{2}(\overline{A B}+\overline{E F})=\frac{1}{2}(13+y)
$$

Thus, we obtain the system

$$
\left\{\begin{array}{l}
y=\frac{1}{2}(x+4) \\
x=\frac{1}{2}(13+y)
\end{array}\right.
$$

from which we find that $x=10, y=7$.