The length of a side of the square if the extermities of its one diagonal are `(1,-2,3)` and `(2,-3,5)` is

Options

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

Answer

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

### Step 1: Identify the coordinates of the extremities of the diagonal.
The extremities of the diagonal are given as:
- Point A: (1, -2, 3)
- Point B: (2, -3, 5)

### Step 2: Use the distance formula to calculate the length of the diagonal.
The distance \(d\) between two points in 3D space \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the coordinates of points A and B:
\[ d = \sqrt{(2 - 1)^2 + (-3 + 2)^2 + (5 - 3)^2} \]
\[ = \sqrt{(1)^2 + (-1)^2 + (2)^2} \]
\[ = \sqrt{1 + 1 + 4} = \sqrt{6} \]

### Step 3: Relate the diagonal length to the side length of the square.
For a square, the relationship between the length of the diagonal \(D\) and the length of a side \(s\) is given by:
\[ D = s\sqrt{2} \]

### Step 4: Solve for the length of a side of the square.
From the relationship, we can express the length of the side \(s\) as:
\[ s = \frac{D}{\sqrt{2}} \]
Substituting \(D = \sqrt{6}\):
\[ s = \frac{\sqrt{6}}{\sqrt{2}} = \sqrt{\frac{6}{2}} = \sqrt{3} \]

### Final Answer:
The length of a side of the square is \(\sqrt{3}\).

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