To find the equation of the side of triangle DEF that is parallel to segment BC of triangle ABC, we need to determine the slope of segment BC first, and then we will use that slope to write the equation of the parallel line through one of the points in triangle DEF.
Step 1: Find the slope of segment BC
To find the slope \( m \) of a line that goes through points \( B (-2, 4) \) and \( C (1, 1) \), we use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where \( (x_1, y_1) = B(-2, 4) \) and \( (x_2, y_2) = C(1, 1) \). Plugging in the coordinates:
\[ m = \frac{1 - 4}{1 - (-2)} = \frac{-3}{1 + 2} = \frac{-3}{3} = -1 \]
Step 2: Find the equation of a line parallel to BC
Since triangle DEF has a side that is parallel to segment BC, it will have the same slope of -1. We need to find a point on triangle DEF to determine the equation of the line.
Let’s choose point \( E(4, 4) \) from triangle DEF.
Step 3: Use point-slope form to find the equation
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Substituting \( m = -1 \) and the coordinates of point \( E(4, 4) \):
\[ y - 4 = -1(x - 4) \]
Simplifying this equation:
\[ y - 4 = -1x + 4 \]
\[ y = -x + 4 + 4 \]
\[ y = -x + 8 \]
Step 4: Write the final equation
Thus, the equation of the side of triangle DEF that is parallel to segment BC in slope-intercept form is:
\[ \boxed{y = -x + 8} \]