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When solving geometry problems such as this one involving hexagonal pyramids, the initial step is to thoroughly understand the problem statement. Breaking down the components helps simplify how we approach the solution. Here, we're dealing with a regular hexagonal pyramid. This means each side of the hexagonal base is equal, specifically 5 cm as provided.
Additionally, the distance from the base of the pyramid to its vertex is given as 15 cm, which is known as the pyramid's altitude. We are tasked with finding the distance from a parallel section to the vertex. This intersection creates a new shape, but because it is parallel to the base, it retains proportional similarities. Understanding this setup is crucial in knowing which properties and formulas can be applied. Key points in tackling such problems:

• Identify and understand all given measurements.
• Visualize the geometric shape and its sectional properties.
• Determine what needs to be found and relate it to the problem's context.

Utilizing these strategies helps systematically approach similar geometry problems.

The concept of geometric similarity plays a vital role in problems involving shapes like a hexagonal pyramid. When a plane cuts the pyramid parallel to its base, the resulting section is similar to the base. This means the shape remains proportionally identical, even though its size differs.
In this exercise, the similarity is leveraged to determine the distances and areas. The areas of similar figures have a specific proportional relationship: the square of the linear scale factor. If the cross-section area and the original base area ratio are found, this ratio squared represents the proportion of sides (or heights). Since the cross-section has been reduced to an area of \( \frac{2}{3} \sqrt{3} \mathrm{~cm}^2 \), compared to the base's \( \frac{75\sqrt{3}}{2} \mathrm{~cm}^2 \), this property allows us to define their size relationship.
To solve:

• Calculate the area ratio \( \frac{\frac{2}{3} \sqrt{3}}{\frac{75\sqrt{3}}{2}} = \frac{4}{225} \).
• Recognize that this represents the square of the linear dimension ratio \( \left(\frac{d}{15}\right)^2 \).
• Solve for \( d \) to find how far the plane is from the pyramid's base, leading to the original question's solution.

Geometric similarity simplifies calculations by providing a direct relationship between dimensions and their squared ratios.

A core formula in this context is the calculation of a regular hexagon's area. Recognizing that a hexagon can be broken down into six equilateral triangles, each sharing the same side length simplifies understanding. For this pyramid's base with a side \( s = 5 \) cm, the formula for a hexagon's area is \( A = \frac{3\sqrt{3}}{2} s^2 \).
Using this, we calculate the base area:

• Plug in the side length: \( \frac{3\sqrt{3}}{2} \times (5)^2 = \frac{75\sqrt{3}}{2} \mathrm{~cm}^2 \).
• This result is fundamental for understanding the proportional changes as the pyramid is intersected by the plane.

Given its regularity, every side and angle aligns evenly, ensuring that each formed triangle contributes equally to the total area. Recognizing and utilizing formulas like this is critical for extrapolating needed dimensions and further solving intersection or volume problems in geometry. Thus, mastering these calculations expands a student's toolkit for tackling more complex geometric analysis.