If the extremities of the diagonal of a square are
(1, -2, 3) and (3, -4, 3), the the length of the side is

Options

A. `2sqrt(2)` units
B. 2 units
C. `sqrt(2)` units
D. 4 units

Answer

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

### Step 1: Identify the coordinates of the diagonal endpoints
The coordinates of the extremities of the diagonal are given as:
- Point A: \( (1, -2, 3) \)
- Point C: \( (3, -4, 3) \)

### Step 2: Use the distance formula to find the length of the diagonal
The distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in three-dimensional space 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 C into the formula:
\[ d = \sqrt{(3 - 1)^2 + (-4 + 2)^2 + (3 - 3)^2} \]
Calculating each component:
- \( (3 - 1)^2 = 2^2 = 4 \)
- \( (-4 + 2)^2 = (-2)^2 = 4 \)
- \( (3 - 3)^2 = 0^2 = 0 \)

Now, substituting these values back into the distance formula:
\[ d = \sqrt{4 + 4 + 0} = \sqrt{8} = 2\sqrt{2} \]

### 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 side length \( a \) is given by:
\[ d = a\sqrt{2} \]
From our calculation, we have \( d = 2\sqrt{2} \).

### Step 4: Solve for the side length \( a \)
Setting the two expressions for the diagonal equal to each other:
\[ 2\sqrt{2} = a\sqrt{2} \]
Dividing both sides by \( \sqrt{2} \):
\[ 2 = a \]

### Conclusion
The length of the side of the square is \( 2 \) units.

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