The problem presented involves a cube with specific dimensions and asks for the distances between various points. Here's a breakdown of how to solve it.

Given:

1. Cube ABCD.EFGH with edge length = 6 cm.
2. Point P is the midpoint of edge EH.
3. Point Q is the intersection of diagonals EG and FH.

Tasks:

a. Find the distance between C and E.
b. Find the distance between C and P.
c. Find the distance between B and Q.

Solutions:

a. Distance between C and E:

Points C and E are vertices of the cube. Since they lie on diagonally opposite ends of the cube, their distance is the cube's space diagonal.

The formula for the space diagonal of a cube is: $\text{Space diagonal} = \sqrt{a^2 + a^2 + a^2} = a\sqrt{3}$ where $a = 6$ cm is the edge length.

$\text{Distance (C to E)} = 6\sqrt{3} \, \text{cm}$

b. Distance between C and P:

Point P is the midpoint of edge EH. To find its coordinates:

• Assume the cube is positioned in 3D space, with A(0, 0, 0), B(6, 0, 0), C(6, 6, 0), D(0, 6, 0), E(0, 0, 6), F(6, 0, 6), G(6, 6, 6), and H(0, 6, 6).
• Edge EH connects E(0, 0, 6) and H(0, 6, 6).
• Midpoint of EH: $P = \left(\frac{0 + 0}{2}, \frac{0 + 6}{2}, \frac{6 + 6}{2}\right) = (0, 3, 6)$

To find the distance between C(6, 6, 0) and P(0, 3, 6), use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ $d = \sqrt{(0 - 6)^2 + (3 - 6)^2 + (6 - 0)^2} = \sqrt{36 + 9 + 36} = \sqrt{81} = 9 \, \text{cm}$

c. Distance between B and Q:

Point Q is the intersection of the cube's body diagonals EG and FH. In a cube, the intersection of the body diagonals is the cube's center.

The center of the cube (and coordinates of Q) is: $Q = \left(\frac{0 + 6}{2}, \frac{0 + 6}{2}, \frac{0 + 6}{2}\right) = (3, 3, 3)$

To find the distance between B(6, 0, 0) and Q(3, 3, 3): $d = \sqrt{(3 - 6)^2 + (3 - 0)^2 + (3 - 0)^2}$ $d = \sqrt{(-3)^2 + 3^2 + 3^2} = \sqrt{9 + 9 + 9} = \sqrt{27} = 3\sqrt{3} \, \text{cm}$

Final Answers:

a. $6\sqrt{3}$ cm
b. $9$ cm
c. $3\sqrt{3}$ cm

Would you like further clarification or a visualization of these points and calculations?

5 Related Questions:

1. How can the formula for the space diagonal of a cube be derived geometrically?
2. What is the distance formula in 3D space, and how does it extend from 2D?
3. How would the solution change if the cube's edge length were different?
4. What is the geometric significance of the point Q in the cube?
5. How can vectors be used to find distances in 3D space?

Tip:

Always sketch a cube and label the points to better understand spatial relationships when solving 3D geometry problems!