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? (1 point)
Responses

x-axis
x-axis

y-axis
y-axis

y = x
y = x

y = -1

Answer 1

To find the line of reflection that transforms the coordinate \( Y(-3, 2) \) into \( Y'(2, -3) \), we can determine the midpoint between the original point and its image. The line of reflection will be the perpendicular bisector of the segment connecting these two points.

**Step 1: Calculate the midpoint.**

The midpoint \( M \) of points \( (-3, 2) \) and \( (2, -3) \) is given by:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + 2}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{-1}{2}, \frac{-1}{2} \right)
\]

So, the midpoint \( M \) is \( \left( -\frac{1}{2}, -\frac{1}{2} \right) \).

**Step 2: Find the slope of line segment \( YY' \).**

The slope \( m \) from \( Y \) to \( Y' \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{2 - (-3)} = \frac{-5}{5} = -1
\]

**Step 3: Find the slope of the perpendicular bisector.**

The slope of the perpendicular bisector will be the negative reciprocal of \( -1 \), which is \( 1 \).

**Step 4: Write the equation of the line.**

Using the point-slope form of a line:

\[
y - y_1 = m(x - x_1)
\]

Using the midpoint \( M\left(-\frac{1}{2}, -\frac{1}{2}\right) \) and slope \( 1 \):

\[
y + \frac{1}{2} = 1\left( x + \frac{1}{2} \right)
\]

This simplifies to:

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

Thus, the line of reflection is:

\[
\boxed{y = x}
\]