5. In triangle $\triangle A B C$, segments parallel to side $\overline{A B}$ are drawn from points $X$ and $Y$ on side $\overline{A C}$. Point $X$ divides segment $\overline{A C}$ into two parts such that $|A X|:|X C|=1: 4$. If the areas of the two shaded regions of triangle $\triangle A B C$ are equal, calculate the ratio $|C Y|:|Y A|$.

![](https://cdn.mathpix.com/cropped/2024_06_03_2ffb0beb69bfdbbfa788g-1.jpg?height=454&width=528&top_left_y=1960&top_left_x=844)

Each problem is scored out of 10 points.

The use of a pocket calculator or any reference materials is not allowed.
5. Let $D$ be the second endpoint of the line segment parallel to segment $\overline{A B}$ through point $X$, and $E$ be the second endpoint of the line segment parallel to segment $\overline{A B}$ through point $Y$.

![](https://cdn.mathpix.com/cropped/2024_05_30_ad5ee125a5da4d07a28fg-3.jpg?height=445&width=511&top_left_y=2133&top_left_x=795)

We observe triangles $\triangle X D C$ and $\triangle A B C$. It is given that $|\angle X C D|=|\angle A C B|$ (common angle at vertex $C$) and $|\angle C X D|=|\angle C A B|$ (angles with parallel sides or angles along the transversal). According to the AA similarity criterion, triangles $\triangle X D C$ and $\triangle A B C$ are similar.

1 POINT From $|C X|:|X A|=4: 1$ we conclude that the ratio $|C X|:|C A|=4: 5$, i.e., the similarity coefficient of these triangles is $k=\frac{4}{5}$.

2 POINTS

The areas of similar triangles are in the ratio of the squares $(k \cdot k)$ of the corresponding side lengths, so we have

$p_{\triangle X D C}: p_{\triangle A B C}=\frac{4}{5} \cdot \frac{4}{5}=\frac{16}{25}$, or $p_{\triangle X D C}=\frac{16}{25} p_{\triangle A B C}$.

1 POINT

Express the area of the shaded part, i.e., the area of quadrilateral $A B D X$.

$p_{A B D X}=p_{\triangle A B C}-p_{X D C}=p_{\triangle A B C}-\frac{16}{25} p_{\triangle A B C}=\frac{9}{25} p_{\triangle A B C}$.

1 POINT

From the problem statement, $p_{\triangle Y E C}=p_{A B D X}=\frac{9}{25} p_{\triangle A B C}$.

1 POINT

Triangles $\triangle Y E C$ and $\triangle A B C$ are also similar according to the AA similarity criterion (common angle at vertex $C$ and congruent angles along the transversal).

From the ratio of the areas of these triangles $p_{\triangle Y E C}: p_{\triangle A B C}=9: 25$, it follows that the similarity coefficient is $k=|C Y|:|C A|=3: 5$.