The length of the perpendicular from the origin to the plane `3x + 4y + 12 z =52` is

Answer

To find the length of the perpendicular from the origin to the plane given by the equation \(3x + 4y + 12z = 52\), we can use the formula for the distance from a point to a plane. The formula is:

\[ \text{Distance} = \frac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}} \]

where \( (x_0, y_0, z_0) \) are the coordinates of the point, and \( ax + by + cz = d \) is the equation of the plane.

### Step 1: Identify the coefficients and the point
From the equation of the plane \(3x + 4y + 12z = 52\), we can identify:
- \(a = 3\)
- \(b = 4\)
- \(c = 12\)
- \(d = 52\)

The point from which we want to calculate the distance is the origin, which has coordinates:
- \( (x_0, y_0, z_0) = (0, 0, 0) \)

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

\[ \text{Distance} = \frac{|3(0) + 4(0) + 12(0) - 52|}{\sqrt{3^2 + 4^2 + 12^2}} \]

### Step 3: Simplify the numerator
Calculating the numerator:

\[ 3(0) + 4(0) + 12(0) - 52 = 0 - 52 = -52 \]

Taking the absolute value:

\[ |-52| = 52 \]

### Step 4: Simplify the denominator
Now, calculate the denominator:

\[ \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \]

### Step 5: Calculate the distance
Now we can calculate the distance:

\[ \text{Distance} = \frac{52}{13} = 4 \]

### Conclusion
The length of the perpendicular from the origin to the plane \(3x + 4y + 12z = 52\) is \(4\) units.

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