The perpendicular distance of the point `(6,5,8)` from y-axis is

Options

A. 5 units
B. 6 units
C. 8 units
D. 10 units

Explanation

To find the perpendicular distance of the point \( (6, 5, 8) \) from the y-axis, we can follow these steps:

### Step 1: Understand the Coordinates
The point \( (6, 5, 8) \) has the following coordinates:
- \( x = 6 \)
- \( y = 5 \)
- \( z = 8 \)

The y-axis is defined by the points where \( x = 0 \) and \( z = 0 \). Therefore, any point on the y-axis can be represented as \( (0, y, 0) \).

### Step 2: Identify the Closest Point on the y-axis
The closest point on the y-axis to the point \( (6, 5, 8) \) would be \( (0, 5, 0) \). This is because the y-coordinate remains the same (5), while the x and z coordinates become 0.

### Step 3: Use the Distance Formula
The distance \( d \) between the point \( (6, 5, 8) \) and the point \( (0, 5, 0) \) can be calculated using the distance formula in three-dimensional space:

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

Here, we can assign:
- \( (x_1, y_1, z_1) = (6, 5, 8) \)
- \( (x_2, y_2, z_2) = (0, 5, 0) \)

### Step 4: Substitute the Coordinates into the Formula
Substituting the coordinates into the distance formula gives:

\[ d = \sqrt{(0 - 6)^2 + (5 - 5)^2 + (0 - 8)^2} \]

### Step 5: Simplify the Expression
Calculating each term:

\[ d = \sqrt{(-6)^2 + (0)^2 + (-8)^2} \]
\[ d = \sqrt{36 + 0 + 64} \]
\[ d = \sqrt{100} \]

### Step 6: Calculate the Final Distance
Taking the square root:

\[ d = 10 \]

Thus, the perpendicular distance of the point \( (6, 5, 8) \) from the y-axis is **10 units**.

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