Find the length of perpendicular from origin to the line `x+7y+4sqrt(2)=0`.

Answer

To find the length of the perpendicular from the origin to the line given by the equation \( x + 7y + 4\sqrt{2} = 0 \), we will use the formula for the distance \( d \) from a point \( (x_1, y_1) \) to a line given by the equation \( Ax + By + C = 0 \):

\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

### Step-by-Step Solution:

1. **Identify the coefficients**:
From the line equation \( x + 7y + 4\sqrt{2} = 0 \), we can identify:
- \( A = 1 \)
- \( B = 7 \)
- \( C = 4\sqrt{2} \)

2. **Identify the point**:
We need to find the distance from the origin, which has coordinates \( (x_1, y_1) = (0, 0) \).

3. **Substitute into the distance formula**:
Substitute \( A \), \( B \), \( C \), \( x_1 \), and \( y_1 \) into the distance formula:
\[ d = \frac{|1 \cdot 0 + 7 \cdot 0 + 4\sqrt{2}|}{\sqrt{1^2 + 7^2}} \]

4. **Calculate the numerator**:
The numerator simplifies to:
\[ |0 + 0 + 4\sqrt{2}| = |4\sqrt{2}| = 4\sqrt{2} \]

5. **Calculate the denominator**:
The denominator simplifies to:
\[ \sqrt{1^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2} \]

6. **Combine the results**:
Now, substitute the values back into the formula:
\[ d = \frac{4\sqrt{2}}{5\sqrt{2}} \]

7. **Simplify**:
The \( \sqrt{2} \) in the numerator and denominator cancels out:
\[ d = \frac{4}{5} \]

### Final Answer:
The length of the perpendicular from the origin to the line \( x + 7y + 4\sqrt{2} = 0 \) is \( \frac{4}{5} \).