Find the equation to and the length of the perpendicular drawn from the point(1,1) upon

the straight line 3x + 4y + 5 = 0. The angle between the axis being 120°.

To find the equation of the perpendicular line from point (1,1) to the given line, we first need to find the slope of the given line.

The equation of the given line is 3x + 4y + 5 = 0. We can rewrite this equation in slope-intercept form (y = mx + c) by solving for y:

4y = -3x - 5
y = (-3/4)x - 5/4

Comparing this equation with y = mx + c, we can see that the slope (m) of the given line is -3/4.

To find the slope of the perpendicular line, we take the negative reciprocal of the slope of the given line. The negative reciprocal of -3/4 is 4/3.

Now, we have the slope (m) of the perpendicular line, and we also have a point (1,1) that lies on this line. We can use the point-slope form (y - y1 = m(x - x1)) to find the equation of the perpendicular line:

y - 1 = (4/3)(x - 1)

Expanding and rearranging the equation, we get:

3y - 3 = 4x - 4

Rearranging further, we obtain the equation of the perpendicular line:

4x - 3y + 1 = 0

Now, let's find the length of the perpendicular line drawn from point (1,1) to the given line.

To calculate the length of the perpendicular line, we need to find the distance between the point (1,1) and the intersection point of the given line and the perpendicular line.

We can solve the system of equations consisting of the given line (3x + 4y + 5 = 0) and the equation of the perpendicular line (4x - 3y + 1 = 0) to find the intersection point.

By substituting the equation of the perpendicular line into the given line, we have:

3x + 4((4/3)x - (4/3)) + 5 = 0

Simplifying the equation, we get:

3x + (16/3)x - 16/3 + 5 = 0

Combining like terms, we have:

(25/3)x - 1/3 = 0

To solve for x, we can cross multiply:

25x - 1 = 0

25x = 1

x = 1/25

Substituting this value of x into the equation of the perpendicular line, we have:

4(1/25) - 3y + 1 = 0

4/25 - 3y + 1 = 0

3y = 4/25 + 1 - 4/25

3y = 51/25

y = 51/75

y = 17/25

Therefore, the intersection point of the given line and the perpendicular line is (1/25, 17/25).

Now, we can calculate the distance between the point (1,1) and the intersection point using the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the values, we get:

d = sqrt((1/25 - 1)^2 + (17/25 - 1)^2)

Simplifying:

d = sqrt((-24/25)^2 + (-8/25)^2)

d = sqrt(576/625 + 64/625)

d = sqrt(640/625)

d = sqrt(64/625)

Simplifying further, we have:

d = 8/25

Therefore, the length of the perpendicular line drawn from the point (1,1) to the given line is 8/25.