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The question asks to find the ratio of the total height of the tent to the height of the door based on the diagram. The tent is described as an equilateral triangle with a square door whose top corners touch the sides of the triangle.

### Explanation:

1. **Equilateral Triangle Properties**:
   - Each side is equal, and the altitude (height) can be calculated using the formula:  
     \[
     h = \frac{\sqrt{3}}{2} \times s
     \]
     where \(s\) is the length of a side of the triangle, and \(\sqrt{3} = 1.732\) is given.

2. **Door Structure**:
   - The door is a square, meaning its height equals its width. Let the height of the door be denoted as \( h_{\text{door}} \).

3. **Height of the Triangle Above the Door**:
   - The height from the top of the door to the apex of the triangle forms a smaller equilateral triangle. The length of this side is reduced since the base of the triangle does not extend below the door.

4. **Finding the Total Height in Relation to the Door**:
   - From the diagram, the total height of the tent includes both the height of the door and the height above it. If we focus on proportions, we find that the total height is \( \frac{\sqrt{3}}{2} \) times the height of the door.

### Final Answer:
\[
\text{Total height of the tent} = 2.732 \times \text{height of the door}
\]

Thus, the missing value is \(2.732\). Would you like more details or have any questions?

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Here are 5 related questions for better understanding:

1. How is the height of an equilateral triangle calculated?
2. Why is the square root of 3 used in this solution?
3. What properties of a square are relevant in solving this problem?
4. How do proportions work when calculating the height of a triangle with a smaller square inside?
5. Can this method be applied to non-equilateral triangles?

**Tip**: In geometry problems involving regular shapes (like equilateral triangles), symmetry can simplify calculations significantly.

Calculate the ratio of the total height of the tent to the height of the door in the given geometric tent design, where the tent is shaped as an equilateral triangle with a square door at the base.

Discover how to calculate the height of an equilateral triangle relative to a square door based on their geometric properties. This problem utilizes formulas for equilateral triangles and explores proportions to find the total height of a tent design suitable for high school students.

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