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?
Responses
1) x-axis
2) y-axis
3) y=x
4) y=-1

Answer 1

To find the line of reflection that maps the point \( Y(-3, 2) \) to \( Y' (2, -3) \), we can use the property that the midpoint of segment \( YY' \) lies on the line of reflection. Additionally, the line of reflection is perpendicular to the segment connecting the pre-image and the image.

1. First, let's calculate the midpoint \( M \) of \( Y \) and \( Y' \):

\[
M = \left( \frac{-3 + 2}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{-1}{2}, \frac{-1}{2} \right) = \left( -\frac{1}{2}, -\frac{1}{2} \right)
\]

2. Now, we need to determine the slope of the line segment connecting \( Y \) to \( Y' \):

\[
\text{slope of } YY' = \frac{-3 - 2}{2 - (-3)} = \frac{-5}{5} = -1
\]

3. The slope of the line of reflection is the negative reciprocal, which is:

\[
\text{slope of the reflection line} = 1
\]

4. Using the midpoint \( M(-\frac{1}{2}, -\frac{1}{2}) \) and the calculated slope of 1, we can write the equation of the line in point-slope form:

\[
y - (-\frac{1}{2}) = 1(x - (-\frac{1}{2}))
\]

This simplifies to:

\[
y + \frac{1}{2} = x + \frac{1}{2}
\]

Subtracting \( \frac{1}{2} \) from both sides gives us:

\[
y = x
\]

Thus, the line of reflection is \( y = x \).

Hence, the answer is:

**3) y = x**