The text in the image is written in Russian and contains a geometry problem. Let's break it down and solve the problem:

Problem Statement (Translated):

Given: an equilateral triangle with a height of $h = 2\sqrt{3} \text{ cm}$. Find the area of the triangle, if its height divides the base in the ratio 1:2.

Solution:

1. Understanding the problem:

   • The triangle is equilateral, meaning all sides are equal.
   • The height $h = 2\sqrt{3}$ cm.
   • The height divides the base in the ratio 1:2.

2. Finding the side length: In an equilateral triangle, the height divides the base into two equal parts. Since the problem states the height divides the base in the ratio 1:2, we can infer that the full length of the base corresponds to the sum of these parts. However, this is contradictory since in an equilateral triangle, the height should divide the base into equal segments. The correct interpretation would be that the problem means to consider the height only and find the area based on that.

   The height $h$ in an equilateral triangle is given by: $h = \frac{\sqrt{3}}{2} \times a$ where $a$ is the side length of the triangle.

   Substituting the given height: $2\sqrt{3} = \frac{\sqrt{3}}{2} \times a$ Solve for $a$: $a = \frac{2\sqrt{3}}{\frac{\sqrt{3}}{2}} = 4 \text{ cm}$

3. Calculating the area: The area $A$ of an equilateral triangle is given by: $A = \frac{\sqrt{3}}{4} \times a^2$ Substituting $a = 4$ cm: $A = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \text{ cm}^2$

Final Answer:

The area of the equilateral triangle is $\boxed{4\sqrt{3} \text{ cm}^2}$.

Would you like more details or have any questions?

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Relative Questions:

1. How do you calculate the height of an equilateral triangle given its side length?
2. What are the properties of an equilateral triangle?
3. How can the area of a triangle be calculated using different methods?
4. What is the significance of the height in determining the area of a triangle?
5. How would the problem change if the triangle were not equilateral?

Tip:

Always check the problem statement carefully to ensure that all geometric properties are correctly understood before solving the problem. Misinterpretation can lead to incorrect conclusions.