The problem describes a block (rectangular prism) with vertices $ABCD.EFGH$ where some dimensions are given. Here's the detailed breakdown of the problem based on the information and diagram provided:

• $AB = 2 \, \text{cm}$
• $2BC = 2 \, \text{cm}$ (which means $BC = 1 \, \text{cm}$)
• $2AE = 2 \, \text{cm}$ (which means $AE = 1 \, \text{cm}$)
• The question asks for the length of $AH$.

Step-by-Step Solution:

1. Identify the coordinates of the vertices:

   • Since the block is a rectangular prism, let's assume the origin $A$ is at $(0, 0, 0)$.
   • The vertices will have coordinates:
     • $A(0, 0, 0)$
     • $B(2, 0, 0)$
     • $E(0, 1, 0)$
     • $H(0, 1, 1)$

2. Find the length of $AH$:

   • The distance formula between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
   • Applying the coordinates of $A(0, 0, 0)$ and $H(0, 1, 1)$, we get: $AH = \sqrt{(0 - 0)^2 + (1 - 0)^2 + (1 - 0)^2}$ $AH = \sqrt{0 + 1^2 + 1^2} = \sqrt{2}$ Therefore, the length of $AH$ is $\sqrt{2} \, \text{cm}$.

Answer:

The length of $AH$ is $\sqrt{2} \, \text{cm}$.

Do you want more details or have any questions about this solution?

Here are some related questions for further exploration:

1. What is the distance between $A$ and $F$ in the same block?
2. How do you calculate the volume of this rectangular prism?
3. What is the surface area of the block given these dimensions?
4. Can we find the diagonal of face $ABCD$?
5. How would the length of $AH$ change if the height $AE$ was doubled?

Tip: For any 3D shape, applying the distance formula correctly helps find any diagonal or slant height.