The distance between the points (1, 4, 5) and
(2, 2, 3) is

Options

A. 5 units
B. 4 units
C. 3 units
D. 2 units

Answer

To find the distance between the points \( A(1, 4, 5) \) and \( B(2, 2, 3) \), we can use the distance formula in three-dimensional geometry. The distance \( d \) between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is given by the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 1: Identify the coordinates
Let:
- \( A(1, 4, 5) \) where \( x_1 = 1, y_1 = 4, z_1 = 5 \)
- \( B(2, 2, 3) \) where \( x_2 = 2, y_2 = 2, z_2 = 3 \)

### Step 2: Substitute the coordinates into the formula
Now, substitute the coordinates into the distance formula:

\[ d = \sqrt{(2 - 1)^2 + (2 - 4)^2 + (3 - 5)^2} \]

### Step 3: Calculate the differences
Calculate the differences:

- \( x_2 - x_1 = 2 - 1 = 1 \)
- \( y_2 - y_1 = 2 - 4 = -2 \)
- \( z_2 - z_1 = 3 - 5 = -2 \)

### Step 4: Square the differences
Now, square each of these differences:

\[ d = \sqrt{(1)^2 + (-2)^2 + (-2)^2} \]

This simplifies to:

\[ d = \sqrt{1 + 4 + 4} \]

### Step 5: Add the squared values
Add the squared values:

\[ d = \sqrt{9} \]

### Step 6: Take the square root
Finally, take the square root:

\[ d = 3 \]

Thus, the distance between the points \( (1, 4, 5) \) and \( (2, 2, 3) \) is \( 3 \).

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