Question

After a reflection, pre-image coordinate Y (-3, 2) results in the image coordinate of Y' (2, -3). What is the line of reflection?

Answer 1

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:

1. **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 \( \left( -\frac{1}{2}, -\frac{1}{2} \right) \).

2. **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 \( -1 \).

3. **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:

\[
m_{reflection} = 1
\]

4. **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:

\[
\boxed{y = x}
\]