Let's analyze the question step by step. Here’s a breakdown of the given problem and solution steps:

Problem Overview:

1. Setup:

   • $\triangle ABC$ is an equilateral triangle on a horizontal plane $\Pi$, with each side measuring 20 cm.
   • $B$ is raised vertically to point $D$, so that the angle between $AB$ and $\Pi$ is $60^\circ$.
   • $E$ is the midpoint of $BC$ on the plane $\Pi$.

2. Questions: (a) Find the lengths of $BE$ and $DE$.
   (b) Someone claims that $\angle CDE$ is the angle between the planes $ABC$ and $ADE$. Do you agree? Explain your answer.

Solution Steps:

(a) Finding the lengths of $BE$ and $DE$:

1. Step 1: Calculate $BE$:

   • $E$ is the midpoint of $BC$, so $BE = \frac{1}{2} BC$.
   • Since $\triangle ABC$ is equilateral, $BC = 20 \, \text{cm}$.
   • Therefore, $BE = \frac{1}{2} \times 20 = 10 \, \text{cm}$.

2. Step 2: Calculate $DE$:

   • $B$ is raised vertically to $D$, so $BD = 20 \, \text{cm}$ (same as the side length of the triangle).
   • $E$ lies directly below $B$ on $BC$ in the plane $\Pi$. Since $BE = 10 \, \text{cm}$, $\triangle BDE$ is a right triangle with:
     • $BD = 20 \, \text{cm}$ (vertical),
     • $BE = 10 \, \text{cm}$ (horizontal).
   • Use the Pythagorean theorem in $\triangle BDE$: $DE = \sqrt{BD^2 + BE^2} = \sqrt{20^2 + 10^2} = \sqrt{400 + 100} = \sqrt{500} = 10\sqrt{5} \, \text{cm}.$

(b) Analyze the claim about $\angle CDE$:

• Claim: $\angle CDE$ is the angle between the planes $ABC$ and $ADE$.

• Analysis:

  • The angle between two planes is defined as the angle between their normal vectors.
  • In this problem:
    • The plane $ABC$ is horizontal ($\Pi$).
    • The plane $ADE$ is inclined, as $D$ is above $\Pi$.
  • The line $DE$ lies in both planes $ABC$ and $ADE$, so $DE$ is the line of intersection of the two planes.
  • $\angle CDE$ is indeed the dihedral angle (angle between the two planes along their line of intersection).

• Conclusion: The claim is correct.

Final Answer:

(a) $BE = 10 \, \text{cm}, \, DE = 10\sqrt{5} \, \text{cm}.$
(b) Yes, $\angle CDE$ is the angle between the planes $ABC$ and $ADE$. The explanation is as above.

Let me know if you'd like further clarifications or a more detailed explanation!
Here are 5 related questions to expand understanding:

1. How do you compute the length of a side in an equilateral triangle given its altitude?
2. What is the formula to find the angle between two planes using their normal vectors?
3. If the triangle's side length was doubled, how would $BE$ and $DE$ change?
4. How can you find the area of the plane $ADE$?
5. What happens to the angle $\angle CDE$ if $D$ is lowered closer to $\Pi$?

Tip: Always draw a 3D diagram and mark right angles to visualize problems involving planes and geometry.