The problem involves determining distances from specific points to planes within a cube. Here's how we can approach it:

We are given a cube $ABCD.EFGH$ with a side length of 16 cm. Let’s break down each part of the problem step by step.

1. Distance from point $B$ to plane $ABFE$

• Plane $ABFE$ is one of the faces of the cube containing points $A, B, F, E$.
• Since point $B$ lies on this plane, the distance from $B$ to this plane is 0 cm.

2. Distance from point $B$ to plane $DCGH$

• Plane $DCGH$ is the opposite face of the cube from $ABFE$. It is parallel to the plane $ABFE$.
• The distance between two parallel planes in a cube is the length of the cube’s edge, which is 16 cm. Therefore, the distance from point $B$ to plane $DCGH$ is 16 cm.

3. Distance from point $C$ to plane $BDG$

• Plane $BDG$ passes through points $B, D, G$, forming a diagonal plane of the cube.
• To find the distance from point $C$ to this plane, we calculate the perpendicular distance from point $C$ to the diagonal plane. This involves more advanced geometry, but since the cube is symmetric, the answer typically involves a formula based on the cube's side length. Here, the result is $\frac{16}{\sqrt{3}}$ cm, the perpendicular distance from any point in the cube to a diagonal plane.

4. Distance from point $C$ to plane $BDE$

• Plane $BDE$ passes through points $B, D, E$. This forms another diagonal plane within the cube.
• The distance from point $C$ to this plane can also be calculated geometrically, similar to the previous case, resulting in $\frac{16}{\sqrt{3}}$ cm.

5. Distance from point $C$ to plane $MNO$

• Here, points $M, N, O$ are defined as midpoints of the edges $BC$, $CD$, and $CG$, respectively.
• The plane $MNO$ is determined by these three midpoints. To find the distance from point $C$ to this plane, we again need to calculate the perpendicular distance from $C$ to the plane formed by these midpoints. The calculation involves the symmetry and midpoints of the cube, which leads to a distance of $\frac{16}{2} = 8$ cm.

Summary of Distances:

1. Distance from point $B$ to plane $ABFE$: 0 cm.
2. Distance from point $B$ to plane $DCGH$: 16 cm.
3. Distance from point $C$ to plane $BDG$: $\frac{16}{\sqrt{3}}$ cm.
4. Distance from point $C$ to plane $BDE$: $\frac{16}{\sqrt{3}}$ cm.
5. Distance from point $C$ to plane $MNO$: 8 cm.

Would you like a detailed explanation on any of these steps or calculations?

Related Questions:

1. What is the formula for the distance from a point to a plane in 3D geometry?
2. How can we calculate the distance between two parallel planes in a cube?
3. What is the geometric significance of a diagonal plane in a cube?
4. How does symmetry help in solving 3D geometry problems involving cubes?
5. How would the solution change if the cube had a different side length?

Tip:

When working with geometry in cubes, understanding symmetry and parallel planes can simplify complex distance calculations.