Given:
- The equation of the straight line is 4x + 3y = 12.
- A right triangle is formed by this line with the axes.

To find:
The length of the perpendicular from the origin to the hypotenuse of the right triangle.

Solution:
Step 1: Find the x and y-intercepts of the line
To find the x-intercept, we set y = 0 and solve for x:
4x + 3(0) = 12
4x = 12
x = 3

To find the y-intercept, we set x = 0 and solve for y:
4(0) + 3y = 12
3y = 12
y = 4

So, the x-intercept is (3, 0) and the y-intercept is (0, 4).

Step 2: Find the equation of the hypotenuse
The hypotenuse of the right triangle is the line segment connecting the x-intercept and the y-intercept.

Using the distance formula, the length of the hypotenuse can be found:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((3 - 0)^2 + (0 - 4)^2)
d = sqrt(3^2 + (-4)^2)
d = sqrt(9 + 16)
d = sqrt(25)
d = 5

So, the length of the hypotenuse is 5 units.

Step 3: Find the equation of the line perpendicular to the hypotenuse passing through the origin
The line perpendicular to the hypotenuse passing through the origin will have a slope that is the negative reciprocal of the slope of the hypotenuse.

The slope of the hypotenuse can be found using the formula:
m = (y2 - y1) / (x2 - x1)
m = (4 - 0) / (3 - 0)
m = 4/3

The slope of the line perpendicular to the hypotenuse will be -3/4.

Using the point-slope form of a line, we can find the equation of the line:
y - y1 = m(x - x1)
y - 0 = (-3/4)(x - 0)
y = (-3/4)x

Step 4: Find the intersection point of the line perpendicular to the hypotenuse and the hypotenuse
To find the intersection point, we need to solve the equations of the two lines simultaneously.
Substitute the equation of the line perpendicular to the hypotenuse into the equation of the hypotenuse:
(-3/4)x = 4 - 3y

Simplify the equation:
-3x = 16 - 12y

Solve for x:
x = (-16 + 12y) / 3

Substitute this value of x into the equation of the hypotenuse:
4((-16 + 12y) / 3) + 3y = 12

Simplify the equation:
-64 + 48y + 9y = 36

Combine like terms:
57y = 100

Solve for y:
y