To find the equation of the perpendicular dropped from vertex C to the internal bisector of angle A, we need to follow these steps:

Step 1: Find the equation of the internal bisector of angle A
Step 2: Find the coordinates of the point where the perpendicular line intersects the internal bisector
Step 3: Use the coordinates found in step 2 to find the equation of the perpendicular line

Let's go through each step in detail:

Step 1: Find the equation of the internal bisector of angle A

The internal bisector of angle A divides angle A into two equal angles. To find the equation of the internal bisector, we need to find the slope of the line passing through vertex A and the midpoint of the side BC.

The coordinates of vertex A are (1,1), and the coordinates of vertex B are (4,-2). The midpoint of side BC can be found using the midpoint formula:

Midpoint of BC = ((4+5)/2, (-2+5)/2) = (9/2, 3/2)

The slope of the line passing through vertex A and the midpoint of BC can be found using the formula:

Slope = (y2 - y1) / (x2 - x1)
= (3/2 - 1) / (9/2 - 1)
= (1/2) / (7/2)
= 1/7

The equation of the internal bisector can be written in slope-intercept form as:

y = (1/7)x + c

Step 2: Find the coordinates of the point where the perpendicular line intersects the internal bisector

The perpendicular line dropped from vertex C will intersect the internal bisector at a point. Let's denote the coordinates of this point as (x, y).

Since the line passing through vertex C is perpendicular to the internal bisector, the product of the slopes of the two lines will be -1. The slope of the internal bisector is 1/7, so the slope of the perpendicular line is -7.

The equation of the perpendicular line can be written in slope-intercept form as:

y = -7x + d

Now, we need to find the values of x and y at the point of intersection of the two lines. To do that, we can solve the system of equations formed by the equations of the internal bisector and the perpendicular line.

(1/7)x + c = -7x + d ...(1)

We also know that the point (5,5) lies on the perpendicular line. Substituting these values into equation (1), we get:

(1/7)(5) + c = -7(5) + d
5/7 + c = -35 + d

Simplifying the equation further, we get:

c - d = -35 - 5/7
c - d = -35 - 35/7
c - d = -35 - 5
c - d = -40

So, the equation (1) can be written as:

(1/7)x + (-40) = -7x + d ...(2)

Step 3: Use the coordinates found in step 2 to find the equation of the perpendicular line

Comparing the coefficients of x in equations (1) and (2), we get:

(1