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The problem asks for the ratio between the shaded area and the total area of the figure.

### Breakdown of the figure:
- The figure consists of two overlapping equilateral triangles forming a star, and inside the star, there is a smaller hexagon with alternating shaded and non-shaded triangles.

### Total Area:
The figure has a symmetrical structure composed of 6 smaller triangular regions arranged in a hexagonal shape, along with the outer triangles from the star.

### Shaded Area:
There are 3 shaded triangles within the central hexagon.

### Ratio Calculation:
- Out of the 6 triangles in the central hexagonal part, 3 are shaded, meaning half of the central hexagonal area is shaded.
- The outer part of the star (the larger triangles that form the star shape) is not shaded.

Therefore, the ratio of the shaded area (3 shaded triangles in the hexagon) to the total area (6 triangles in the hexagon plus the area of the outer triangles) needs to be considered. After reviewing the options, the correct ratio corresponds to **\( \frac{1}{4} \)**.

So, the answer is **(a) \( \frac{1}{4} \)**.

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Would you like further details, or do you have any questions? Here are some related questions:

1. How do you calculate the area of an equilateral triangle?
2. What is the significance of symmetry in solving geometry problems like this?
3. How can we calculate the total area of a star shape?
4. What other methods can be used to find the area ratio in complex shapes?
5. How does the area of a regular hexagon compare to that of its inscribed triangles?

**Tip**: When working with symmetrical shapes, identifying repeating patterns helps in calculating areas efficiently.

La razón entre el área sombreada y el área total de la figura es:

Learn how to calculate the ratio between the shaded area and the total area of a complex star-shaped figure. This problem covers geometry concepts, including the calculation of the area of equilateral triangles and the application of ratios and symmetry. Ideal for students in Grades 9-11.

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