To find the line of reflection given the pre-image coordinates \( Y (-3, 2) \) and the image coordinates \( Y' (2, -3) \), we can follow these steps:
-
Find the midpoint of the segment connecting the points \( Y \) and \( Y' \). The midpoint is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting in the coordinates:
\[ M = \left( \frac{-3 + 2}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{-1}{2}, \frac{-1}{2} \right) \]
So, the midpoint \( M \) is \( \left( -\frac{1}{2}, -\frac{1}{2} \right) \).
-
Find the slope of the line connecting \( Y \) and \( Y' \). The slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{2 - (-3)} = \frac{-5}{5} = -1 \]
The line from \( Y \) to \( Y' \) has a slope of \( -1 \).
-
Determine the slope of the line of reflection. The line of reflection is perpendicular to the line connecting \( Y \) and \( Y' \). If the slope of the line is \( -1 \), the slope of the line of reflection will be the negative reciprocal:
\[ m_{reflection} = 1 \]
-
Use the point-slope form of a line to write the equation of the line of reflection. The point-slope form is:
\[ y - y_1 = m (x - x_1) \]
where \( m = 1 \) and \( (x_1, y_1) = \left( -\frac{1}{2}, -\frac{1}{2} \right) \):
\[ y - \left(-\frac{1}{2}\right) = 1 \left(x - \left(-\frac{1}{2}\right)\right) \]
Simplifying this gives:
\[ y + \frac{1}{2} = x + \frac{1}{2} \]
Rearranging:
\[ y = x \]
Thus, the line of reflection is given by the equation:
\[ \boxed{y = x} \]