The question is asking to determine the distance from point $G$ to the midpoint of edge $AB$ of a cube, where the edge length is $a$ cm.

Understanding the Diagram

• The cube has vertices labeled $A, B, C, D, E, F, G, H$.
• $G$ is the topmost corner of the cube (opposite vertex $A$).
• $AB$ is one of the edges on the base of the cube.

Approach:

1. Coordinates of Key Points:

   • Assuming the cube is aligned along the axes:
     • $A = (0, 0, 0)$
     • $B = (a, 0, 0)$
     • $G = (a, a, a)$

2. Midpoint of $AB$:

   • The midpoint $M$ of $AB$ can be found using the midpoint formula: $M = \left( \frac{0 + a}{2}, \frac{0 + 0}{2}, \frac{0 + 0}{2} \right) = \left( \frac{a}{2}, 0, 0 \right)$

3. Distance between $G$ and $M$:

   • The distance formula in 3D is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
   • Substituting the coordinates of $G(a, a, a)$ and $M\left( \frac{a}{2}, 0, 0 \right)$: $d = \sqrt{\left( a - \frac{a}{2} \right)^2 + (a - 0)^2 + (a - 0)^2}$ $d = \sqrt{\left( \frac{a}{2} \right)^2 + a^2 + a^2}$ $d = \sqrt{\frac{a^2}{4} + a^2 + a^2} = \sqrt{\frac{a^2}{4} + 2a^2} = \sqrt{\frac{9a^2}{4}} = \frac{3a}{2}$

Thus, the distance from $G$ to the midpoint of $AB$ is $\frac{3a}{2}$.

The correct answer is option c: $\frac{3a}{2}$ cm.

Would you like further details on any part of the solution?

Related Questions:

1. How do you find the distance between two points in 3D space?
2. What is the formula to find the midpoint of a line segment?
3. How do you generalize the distance formula for higher dimensions?
4. Can this problem be solved using vectors instead of coordinates?
5. What are the practical applications of distance formulas in geometry?

Tip: Always ensure to apply the correct formula when dealing with 3D coordinates, considering all three axes for accurate results.