To determine whether the two triangular prisms are similar and to compare their volumes, we first need to calculate their volumes. The volume \( V \) of a triangular prism can be calculated using the formula:
\[ V = \text{Base Area} \times \text{Height} \]
For triangular prisms with identical shapes, the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions (i.e., the heights, bases, and lengths).
Given dimensions:
- Prism 1: \( H = 27.5, B = 27.5, L = 55 \)
- Prism 2: \( H = 5.5, B = 5.5, L = 11 \)
Finding the corresponding linear dimension ratio:
For the heights: \[ \text{Height ratio} = \frac{27.5}{5.5} = 5 \]
For the bases: \[ \text{Base ratio} = \frac{27.5}{5.5} = 5 \]
For the lengths: \[ \text{Length ratio} = \frac{55}{11} = 5 \]
Since all ratios are equal and equal to 5, the prisms are similar.
Calculating the volumes:
To find the volume, we first need the area of the triangular base. Let's assume the base is a right triangle for simplicity. However, the area of the base triangle is not given in the question. We can directly find the volume ratio, knowing that both prisms are triangular prisms.
Since the volumes will have a ratio that is the cube of the length (or height) ratio:
\[ \text{Volume ratio} = \left( \text{Ratio of Linear Dimensions} \right)^3 = 5^3 = 125 \]
Conclusion:
The correct statement based on our calculations is that:
The two figures are similar because the volumes have a ratio of 125 : 1.