The distance between the point `P(2m,3m,4m)` and the `x`-axis

Options

A. `sqrt(29)m`
B. `5m`
C. `sqrt(13)m`
D. `sqrt(20)m`

Answer

To find the distance between the point \( P(2m, 3m, 4m) \) and the x-axis, we can follow these steps:

### Step 1: Understand the Coordinates
The coordinates of point \( P \) are given as \( (2m, 3m, 4m) \). The x-axis can be represented by any point of the form \( (x, 0, 0) \), where \( x \) can be any value.

### Step 2: Identify the Perpendicular Distance
The shortest distance from point \( P \) to the x-axis is the perpendicular distance. This distance can be calculated by considering the y and z coordinates of point \( P \) since the x-coordinate does not affect the distance to the x-axis.

### Step 3: Calculate the Distance
The coordinates of the point on the x-axis that is directly below point \( P \) would be \( (2m, 0, 0) \). The distance \( D \) between the point \( P(2m, 3m, 4m) \) and the point \( (2m, 0, 0) \) can be calculated using the distance formula:

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

Substituting the coordinates:

\[ D = \sqrt{(2m - 2m)^2 + (3m - 0)^2 + (4m - 0)^2} \]

This simplifies to:

\[ D = \sqrt{0 + (3m)^2 + (4m)^2} \]

Calculating the squares:

\[ D = \sqrt{0 + 9m^2 + 16m^2} = \sqrt{25m^2} \]

Taking the square root gives:

\[ D = 5m \]

### Final Answer
The distance between the point \( P(2m, 3m, 4m) \) and the x-axis is \( 5m \).

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