Let’s analyze the given question.

Problem Breakdown:

We are given a triangular prism where:

• $[BA] \perp [AC]$, indicating a right triangle at $A$,
• $DE = 6 \, \text{cm}, \, BE = 7 \, \text{cm}, \, EF = 10 \, \text{cm}$, which form the base triangle ($\triangle DEF$),
• The prism height is $7 \, \text{cm}$ as the distance between $AB$ and $DE$ or $AC$ and $EF$.

The goal is to calculate the total surface area of the prism.

Step 1: Area of the Base Triangle ($\triangle DEF$)

Using the triangle area formula: $\text{Area of } \triangle DEF = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times EF \times DE$ Substitute the given values: $\text{Area of } \triangle DEF = \frac{1}{2} \times 10 \times 6 = 30 \, \text{cm}^2$

Step 2: Surface Area of the Prism

The total surface area of a prism includes:

1. The areas of two triangular bases ($2 \times \text{Area of } \triangle DEF$),
2. The areas of three rectangular faces.

Rectangular Faces:

The prism has three rectangular lateral faces, and their areas are calculated as:

• Face 1: $BE \times \text{height} = 7 \times 7 = 49 \, \text{cm}^2$,
• Face 2: $DE \times \text{height} = 6 \times 7 = 42 \, \text{cm}^2$,
• Face 3: $EF \times \text{height} = 10 \times 7 = 70 \, \text{cm}^2$.

Total Surface Area:

Combine all the areas: $\text{Total Surface Area} = 2 \times \text{Area of } \triangle DEF + \text{Sum of Rectangular Faces}$ Substitute the values: $\text{Total Surface Area} = 2 \times 30 + (49 + 42 + 70) = 60 + 161 = 221 \, \text{cm}^2$

Final Answer:

The total surface area of the prism is: $\boxed{221 \, \text{cm}^2}$

Would you like further details or have additional questions?

Follow-Up Questions:

1. How do you calculate the volume of this triangular prism?
2. What is the relationship between the base area and the total surface area?
3. Can you explain why $[BA] \perp [AC]$ affects the geometry of the prism?
4. How do you determine the slant heights for triangular prisms?
5. How does this problem change if the height of the prism is increased?

Tip:

When solving prism-related problems, always start by identifying and calculating the areas of the bases and lateral faces separately for clarity.