Find the length of the perpendicular drawn from
the point `(-3, -4)` upon the straight line
`12 (x+ 6) = 5 (y-2)`.

Answer

To find the length of the perpendicular drawn from the point \((-3, -4)\) to the straight line given by the equation \(12(x + 6) = 5(y - 2)\), we will follow these steps:

### Step 1: Convert the line equation to the standard form
We start with the equation:
\[ 12(x + 6) = 5(y - 2) \]
Expanding both sides:
\[ 12x + 72 = 5y - 10 \]
Rearranging this to the form \(Ax + By + C = 0\):
\[ 12x - 5y + 82 = 0 \]
Here, \(A = 12\), \(B = -5\), and \(C = 82\).

### Step 2: Use the formula for the length of the perpendicular
The formula for the length \(d\) of the perpendicular from a point \((x_1, y_1)\) to the line \(Ax + By + C = 0\) is given by:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \(A = 12\), \(B = -5\), \(C = 82\), \(x_1 = -3\), and \(y_1 = -4\):
\[ d = \frac{|12(-3) + (-5)(-4) + 82|}{\sqrt{12^2 + (-5)^2}} \]

### Step 3: Calculate the numerator
Calculating the numerator:
\[ 12(-3) = -36 \]
\[ -5(-4) = 20 \]
Adding these values with \(C\):
\[ -36 + 20 + 82 = -36 + 102 = 66 \]
Thus, the numerator becomes:
\[ |66| = 66 \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Step 5: Calculate the length of the perpendicular
Now substituting the values back into the formula:
\[ d = \frac{66}{13} \]

### Final Answer
The length of the perpendicular drawn from the point \((-3, -4)\) to the line is:
\[ \frac{66}{13} \]