Find the distance between the points `(2,3,5) and (4,3,1)`.

Options

A. `2 sqrt5`
B. `3 sqrt5`
C. `5 sqrt2`
D. None of these

Answer

To find the distance between the points \( (2, 3, 5) \) and \( (4, 3, 1) \), we can use the distance formula for three-dimensional space. The formula is given by:

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

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

### Step 2: Substitute the coordinates into the distance formula
We substitute the coordinates into the formula:

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

### Step 3: Calculate each term
Now, we calculate each term inside the square root:

1. \( (4 - 2)^2 = 2^2 = 4 \)
2. \( (3 - 3)^2 = 0^2 = 0 \)
3. \( (1 - 5)^2 = (-4)^2 = 16 \)

### Step 4: Add the results
Now, we add these results together:

\[ d = \sqrt{4 + 0 + 16} \]

\[ d = \sqrt{20} \]

### Step 5: Simplify the square root
We can simplify \( \sqrt{20} \):

\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]

### Final Answer
Thus, the distance between the points \( (2, 3, 5) \) and \( (4, 3, 1) \) is:

\[ d = 2\sqrt{5} \]

---