The problem involves a cube with a side length of 6 cm. We are asked to determine the distance between several pairs of points on the cube.

Let's analyze the distances for each pair:

1. Distance from A to F:
   Points A and F are located on the same edge of the cube. The distance between A and F is the side length of the cube, which is 6 cm.

2. Distance from B to G:
   Points B and G are located on the diagonal of one of the faces of the cube. To find the distance of the diagonal on a face, we use the Pythagorean theorem: $\text{Diagonal} = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \, \text{cm}$

3. Distance from A to G:
   Points A and G are diagonally opposite in the three-dimensional space of the cube. To calculate this distance, we use the 3D Pythagorean theorem: $\text{Distance} = \sqrt{6^2 + 6^2 + 6^2} = \sqrt{108} = 6\sqrt{3} \, \text{cm}$

4. Distance from A to P:
   Points A and P are located along one of the edges of the cube. Since P is directly above A, the distance from A to P is simply the side length of the cube, which is 6 cm.

5. Distance from O to B:
   The point O is the center of the cube. To find the distance from the center of the cube (O) to any vertex (B), we use half the diagonal of the cube: $\text{Distance} = \frac{\sqrt{6^2 + 6^2 + 6^2}}{2} = \frac{\sqrt{108}}{2} = 3\sqrt{3} \, \text{cm}$

Answers:

• a. A to F: 6 cm
• b. B to G: $6\sqrt{2}$ cm
• c. A to G: $6\sqrt{3}$ cm
• d. A to P: 6 cm
• e. O to B: $3\sqrt{3}$ cm

Would you like a further explanation of any of these points or do you have any questions?

Additional Related Questions:

1. How do you calculate the diagonal of a cube face?
2. How is the 3D Pythagorean theorem applied in space?
3. What is the formula for the distance between two opposite vertices of a cube?
4. How do you find the center of a geometric shape like a cube?
5. What are other properties related to the diagonals of a cube?

Tip: To find distances in a 3D space, always use the Pythagorean theorem in three dimensions: $\sqrt{x^2 + y^2 + z^2}$.