Fill in the blanks
(c) The length of perpendicular from a point (1, 2) to a line `3x + 4y + 5 = 0` is _________.

Answer

To find the length of the perpendicular from the point (1, 2) to the line given by the equation \(3x + 4y + 5 = 0\), we can use the formula for the perpendicular distance from a point to a line.

### Step-by-Step Solution:

1. **Identify the coefficients from the line equation**:
The line is given in the form \(Ax + By + C = 0\). Here, \(A = 3\), \(B = 4\), and \(C = 5\).

2. **Identify the coordinates of the point**:
The point from which we need to find the perpendicular distance is \( (x_1, y_1) = (1, 2) \).

3. **Use the formula for the distance**:
The formula for the perpendicular distance \(d\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]

4. **Substitute the values into the formula**:
- Calculate the numerator:
\[ |Ax_1 + By_1 + C| = |3(1) + 4(2) + 5| = |3 + 8 + 5| = |16| = 16 \]
- Calculate the denominator:
\[ \sqrt{A^2 + B^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

5. **Calculate the distance**:
\[ d = \frac{16}{5} \]

Thus, the length of the perpendicular from the point (1, 2) to the line \(3x + 4y + 5 = 0\) is \(\frac{16}{5}\).

### Final Answer:
The length of perpendicular from the point (1, 2) to the line \(3x + 4y + 5 = 0\) is \(\frac{16}{5}\).

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