In the following figure, there are equilateral triangles $A B C, D B E, I E F$, and $H I G$. The areas of triangles $D B E, I E F$, and $H I G$ are in the ratio $9: 16: 4$. In what ratio are

1. the lengths of segments $H I$ and $I E$,
2. the areas of triangles $A B C$ and $H E C$?

(K. Pazourek)

![](https://cdn.mathpix.com/cropped/2024_04_17_3afc6eb4999c6025c30cg-2.jpg?height=537&width=552&top_left_y=488&top_left_x=752)
For two equilateral triangles with side lengths $a, b$, it holds that the ratio of their areas is $a^{2}: b^{2}$. Conversely, if the ratio of the areas is $a^{2}: b^{2}$, then the ratio of the sides is $a: b$. This statement can be justified by a general rule for two similar figures or by an explicit calculation, where the area of an equilateral triangle with side $a$ is $\frac{1}{4} \sqrt{3} a^{2}$.

1. If the areas of triangles $H I G$ and $I E F$ are in the ratio $4: 16$, then the ratio of their sides $|H I| : |I E|$ will be $2: 4 = 1: 2$.
2. If we choose the unit $j = \frac{1}{2}|H I|$, then $|I E| = |E F| = 4 j$. In the equilateral triangle $H E C$, it holds that $|H I| + |I E| = |E F| + |F C|, |I E| = |E F|$, thus $|F C| = |H I| = 2 j$. From the ratio $9: 16$ of the areas of triangles $D B E$ and $I E F$, it follows that $|B E|:|E F| = 3: 4$, hence

$$
|B E| = \frac{3}{4}|E F| = \frac{3}{4} \cdot 4 j = 3 j
$$

Thus, the side $B C$ of triangle $A B C$ has the length

$$
|B C| = |B E| + |E F| + |F C| = 3 j + 4 j + 2 j = 9 j \text{. }
$$

Similarly, the length of the side $H E$ of triangle $H E C$ is

$$
|H E| = |H I| + |I E| = 2 j + 4 j = 6 j.
$$

The ratio of the sides of triangles $A B C$ and $H E C$ is $9: 6$, which simplifies to $3: 2$. The ratio of the areas of these triangles is $3^{2}: 2^{2} = 9: 4$.

Evaluation. 1 point for the initial consideration; 1 point for the first part of the problem; 1 point each for expressing the sides of triangles $H E C$ and $A B C$; 2 points for calculating the ratio of the areas of triangles $A B C$ and $H E C$. (The final ratios do not need to be in their simplest form.)

Alternative Solution. Divide triangle $A B C$ into equilateral triangles with side length $\frac{1}{2}|H I| = 1 j$. The ratio $9: 16: 4$ of the areas of triangles $D B E, I E F, H I G$ also indicates the ratio of the number of these small triangles in the respective larger triangles. Since triangle $H I G$ is divided into 4 small triangles, then triangles $I E F$ and $D B E$ must consist of 16 and 9 small triangles, respectively.

![](https://cdn.mathpix.com/cropped/2024_04_17_3afc6eb4999c6025c30cg-3.jpg?height=534&width=552&top_left_y=178&top_left_x=752)

From the diagram, the following can be deduced:

1. The ratio $|H I|:|I E|$ is $2 j: 4 j = 1: 2$.
2. The ratio $S_{A B C}: S_{H E C}$ is the same as the ratio of the number of small triangles contained in these triangles, i.e., $81: 36 = 9: 4$.

Evaluation. 1 point for dividing triangles $H I G$ and $I E F$ into small triangles; 1 point for calculating the ratio $|H I|:|I E|$; 2 points for dividing the remaining area of triangle $A B C$ into small triangles; 2 points for calculating the ratio $S_{A B C}: S_{H E C}$. (The final ratios do not need to be in their simplest form.)