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Let's work through this geometry problem step by step. The sheet \( ABCD \) is folded along \( BD \), and the goal is to calculate various lengths and angles as described in the problem.

### (a) Finding the lengths of \( BD \), \( ED \), and \( AE \) (in surd form):
- **Length of \( BD \):**
  Since \( ABCD \) is a square of side 3 meters, the diagonal \( BD \) can be found using the Pythagorean theorem in triangle \( ABD \):
  \[
  BD = \sqrt{AB^2 + AD^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \text{ meters}.
  \]

- **Length of \( ED \):**
  \( E \) is directly below \( B \), and \( BE = 2 \) meters. So, \( E \) is 2 meters vertically below \( B \) along the perpendicular to plane II. Since \( D \) is on the plane, the length \( ED \) can be found using Pythagoras’ theorem in triangle \( BDE \), where \( BD = 3\sqrt{2} \) and \( BE = 2 \):
  \[
  ED = \sqrt{BD^2 - BE^2} = \sqrt{(3\sqrt{2})^2 - 2^2} = \sqrt{18 - 4} = \sqrt{14} \text{ meters}.
  \]

- **Length of \( AE \):**
  \( AE \) is in triangle \( ABE \). We know \( AB = 3 \) meters and \( BE = 2 \) meters. Using the Pythagorean theorem in triangle \( ABE \):
  \[
  AE = \sqrt{AB^2 + BE^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \text{ meters}.
  \]

### (b) Finding \( \angle LADE \):
Here, \( L \) represents the line along the fold \( BD \). We need to calculate \( \angle LADE \), which is the angle between \( AD \) and \( AE \) within the triangle \( ADE \). Since \( AD = 3 \) and \( AE = \sqrt{13} \), and using the cosine rule:
  \[
  \cos(\angle LADE) = \frac{AD^2 + AE^2 - ED^2}{2 \cdot AD \cdot AE} = \frac{3^2 + (\sqrt{13})^2 - (\sqrt{14})^2}{2 \cdot 3 \cdot \sqrt{13}} = \frac{9 + 13 - 14}{6\sqrt{13}} = \frac{8}{6\sqrt{13}}.
  \]
So, \( \angle LADE = \cos^{-1}\left(\frac{4}{3\sqrt{13}}\right) \).

### (c) Angle between \( BD \) and the plane II:
Since \( B \) is 2 meters above the plane, the angle between \( BD \) and the horizontal plane can be calculated using the right triangle formed by \( BDE \). The angle \( \theta \) is between \( BD \) and \( BE \), so:
  \[
  \sin(\theta) = \frac{BE}{BD} = \frac{2}{3\sqrt{2}}.
  \]
Thus, \( \theta = \sin^{-1}\left(\frac{2}{3\sqrt{2}}\right) \).

### (d) Angle between planes \( ABD \) and \( CBD \):
The angle between the planes can be computed by finding the normal vectors to both planes and using the dot product. Plane \( ABD \) involves vectors \( AB \) and \( BD \), while plane \( CBD \) involves vectors \( CB \) and \( BD \). The angle \( \phi \) between these planes is:
  \[
  \cos(\phi) = \frac{\text{Dot product of normals of planes ABD and CBD}}{\text{Magnitude of both normals}}.
  \]
  We'll calculate the vectors and compute \( \phi \), but it involves more detailed vector operations, which require some algebraic steps.

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Would you like me to dive into more details or expand on any step? Here are five related questions:

1. How would you find the vector normal to plane \( ABD \)?
2. Can you compute the precise dot product between the normal vectors?
3. What are the steps to find the exact angle in degrees for part (c)?
4. How does the Pythagorean theorem apply to three-dimensional problems like this?
5. How can the cosine rule be adapted for 3D geometry?

**Tip:** When calculating angles between planes, using vectors and dot products simplifies the process significantly.

In Figure (1), ABCD is a thin square metal sheet of side three meters. The metal sheet is folded along BD, and the edges AD and CD of the folded metal sheet are placed on a horizontal plane II with B two meters vertically above the plane II. E is the foot of the perpendicular from B to the plane II. (See Figure 2.) Find the lengths of BD, ED, and AE. Find ∠LADE. Find the angle between BD and the plane II. Find the angle between the planes ABD and CBD.

Learn how to solve a 3D geometry problem involving a folded metal sheet, using the Pythagorean theorem and cosine rule to find the lengths BD, ED, AE, and the angles between lines and planes. This problem is suitable for high school students, Grades 10-12.

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