If the end points of the diagonal of a square are (-2,3,1) and (-3,5,2) , then the length of the side of the square is

Options

A. ` sqrt(2)` units
B. `sqrt(3)` units
C. ` sqrt(6)` units
D. ` 2 sqrt(3)` units

Answer

To find the length of the side of the square given the endpoints of its diagonal, we can follow these steps:

### Step 1: Identify the endpoints of the diagonal
The endpoints of the diagonal of the square are given as \( A(-2, 3, 1) \) and \( B(-3, 5, 2) \).

### Step 2: Calculate the length of the diagonal
The length of the diagonal \( d \) can be calculated using the distance formula in three-dimensional space:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ d = \sqrt{((-3) - (-2))^2 + (5 - 3)^2 + (2 - 1)^2} \]
\[ = \sqrt{(-3 + 2)^2 + (5 - 3)^2 + (2 - 1)^2} \]
\[ = \sqrt{(-1)^2 + (2)^2 + (1)^2} \]
\[ = \sqrt{1 + 4 + 1} \]
\[ = \sqrt{6} \]

### Step 3: Relate the diagonal length to the side length
Let the length of the side of the square be \( a \). The relationship between the diagonal \( d \) and the side length \( a \) of a square is given by:
\[ d = a\sqrt{2} \]
From the previous step, we have \( d = \sqrt{6} \). Therefore, we can set up the equation:
\[ \sqrt{6} = a\sqrt{2} \]

### Step 4: Solve for the side length \( a \)
To find \( a \), we can rearrange the equation:
\[ a = \frac{\sqrt{6}}{\sqrt{2}} \]
We can simplify this:
\[ a = \sqrt{\frac{6}{2}} = \sqrt{3} \]

### Conclusion
The length of the side of the square is \( \sqrt{3} \) units.

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