LOTS OF POINTS GIVING BRAINLIEST I NEED HELP PLEASEE Triangle ABC is similar to triangle DEF. Write the equation, in slope-intercept form, of the side of triangle DEF that is parallel to segment BC. You must show all work to receive credit.

Triangle ABC and DEF are shown. A is at negative 2, 1. B is at negative 2, 4. C is at 1, 1. D is at 4, 2. E is at 4, 4. F is at 6, 2.

Segment EF: y = -x + 8

Segment BC: y = -x + 2

Step-by-step explanation:

Given the two similar right triangles, ΔABC and ΔDEF, for which we must determine the slope-intercept form of the side of ΔDEF that is parallel to segment BC.

Upon observing the given diagram, we can infer the following corresponding sides:

We must determine the slope of segment BC from ΔABC, which corresponds to segment EF from ΔDEF.

In order to solve for the slope of segment BC, we can use the following slope formula:

Use the following coordinates from the given diagram:

Point B:  (x₁, y₁) =  (-2, 4)

Point C:  (x₂, y₂) = ( 1,  1 )

Substitute these values into the slope formula:

Similar to how we determined the slope of segment BC, we will use the coordinates of points E and F from ΔDEF to find its slope:

Point E:  (x₁, y₁) =  (4, 4)

Point F:  (x₂, y₂) = (6, 2)

Substitute these values into the slope formula:

Our calculations show that segment BC and EF have the same slope of -1.  In geometry, we know that two nonvertical lines are parallel if and only if they have the same slope.  

Since segments BC and EF have the same slope, then it means that  .

The y-intercept is the point on the graph where it crosses the y-axis. Thus, it is the value of "y" when x = 0.

Using the slope of segment BC, m = -1, and the coordinates of point C, (1,  1), substitute these values into the slope-intercept form (y = mx + b) to solve for the y-intercept, b.

y = mx + b

1 = -1( 1 ) + b

1 = -1 + b

Add 1 to both sides to isolate b:

1 + 1 = -1 + 1 + b

2 = b

Hence, the y-intercept of segment BC is: b = 2.

Therefore, the linear equation in slope-intercept form of segment BC is:

⇒  y = -x + 2.

Using the slope of segment EF, m = -1, and the coordinates of point E, (4, 4), substitute these values into the slope-intercept form to solve for the y-intercept, b.

y = mx + b

4 = -1( 4 ) + b

4 = -4 + b

Add 4 to both sides to isolate b:

4 + 4 = -4 + 4 + b

8 = b

Hence, the y-intercept of segment BC is: b = 8.

Therefore, the linear equation in slope-intercept form of segment EF is:

⇒  y = -x + 8.