Given:
The point (1,-2) lies on the line 3y = 4x - 5.

To find:
1. The length of the perpendicular drawn from the point (1,-2) on the line.
2. The coordinates of the foot of the perpendicular.

Explanation:
To find the length of the perpendicular, we need to first find the equation of the line perpendicular to the given line that passes through the point (1,-2). Let's start by finding the slope of the given line.

Step 1: Finding the slope of the given line:
The given equation is 3y = 4x - 5.
We can rewrite this equation in slope-intercept form (y = mx + b) by dividing both sides by 3:
y = (4/3)x - 5/3.
Comparing this equation with the slope-intercept form, we can see that the slope of the given line is 4/3.

Step 2: Finding the slope of the perpendicular line:
The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
Therefore, the slope of the perpendicular line = -1/(4/3) = -3/4.

Step 3: Finding the equation of the perpendicular line:
We have the slope of the perpendicular line and a point it passes through, (1,-2).
Using the point-slope form of a linear equation (y - y1 = m(x - x1)), we can write the equation of the perpendicular line as:
y - (-2) = (-3/4)(x - 1).
Simplifying the equation, we get:
y + 2 = (-3/4)x + 3/4.

Step 4: Finding the coordinates of the foot of the perpendicular:
The foot of the perpendicular is the point where the perpendicular line intersects the given line. To find this point, we need to solve the system of equations formed by the given line and the perpendicular line.

Using the equations:
3y = 4x - 5 (given line)
y + 2 = (-3/4)x + 3/4 (perpendicular line)

Solving these equations, we get the coordinates of the foot of the perpendicular as (7/2, -5/2).

Step 5: Finding the length of the perpendicular:
The length of the perpendicular is the distance between the point (1,-2) and the foot of the perpendicular (7/2, -5/2).
Using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates, we get:
d = sqrt((7/2 - 1)^2 + (-5/2 - (-2))^2)
Simplifying the expression, we get:
d = sqrt(25/4 + 9/4)
d = sqrt(34/4)
d = sqrt(17/2)

Answer:
1. The length of the perpendicular drawn from the point (1,-2) on the line 3y = 4x - 5 is sqrt(17/2).
2. The coordinates of the foot of the perpendicular are (7/2,