Find the perpendicular distance of an angular point of a cube from a diagona which does not pass through that angular point. (a) `(1)/sqrt3` (b) `2/3` (c) `sqrt((2)/(3))` (d) `(sqrt3)/2`

Options

A. `(1)/(sqrt3(`
B. `2/3`
C. `sqrt((2)/(3))`
D. `(sqrt3)/2`

Answer

To find the perpendicular distance of an angular point of a cube from a diagonal that does not pass through that angular point, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Cube:**
Let’s consider a cube with side length \( a \). We can place the cube in a 3D coordinate system such that one vertex (angular point) is at the origin \( O(0, 0, 0) \) and the opposite vertex is at \( A(a, a, a) \).

2. **Identify the Diagonal:**
The diagonal we will consider is from point \( O(0, 0, 0) \) to point \( A(a, a, a) \). The coordinates of these points are:
- \( O = (0, 0, 0) \)
- \( A = (a, a, a) \)

3. **Calculate the Length of the Diagonal:**
The length of the diagonal \( OA \) can be calculated using the distance formula:
\[ OA = \sqrt{(a - 0)^2 + (a - 0)^2 + (a - 0)^2} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]

4. **Find the Midpoint of the Diagonal:**
The midpoint \( M \) of the diagonal \( OA \) is given by:
\[ M = \left( \frac{0 + a}{2}, \frac{0 + a}{2}, \frac{0 + a}{2} \right) = \left( \frac{a}{2}, \frac{a}{2}, \frac{a}{2} \right) \]

5. **Determine the Perpendicular Distance:**
The perpendicular distance from the angular point \( O(0, 0, 0) \) to the diagonal \( OA \) can be found using the formula for the distance from a point to a line in 3D. However, since we know the properties of the cube, we can simplify this:
The distance from the point \( O \) to the line segment \( OA \) is half the length of the diagonal \( OA \):
\[ \text{Perpendicular Distance} = \frac{OA}{2} = \frac{a\sqrt{3}}{2} \]

6. **Substituting \( a = 1 \):**
For a unit cube where \( a = 1 \):
\[ \text{Perpendicular Distance} = \frac{1\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

### Final Answer:
Thus, the perpendicular distance of an angular point of a cube from a diagonal that does not pass through that angular point is:
\[ \frac{\sqrt{3}}{2} \]