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 between a point and its image, we can use the midpoint formula. The midpoint of the line segment connecting the original point \( Y(-3, 2) \) and its image \( Y'(2, -3) \) will lie on the line of reflection.

1. **Find the Midpoint**:
The midpoint \( M \) between the coordinates \( (-3, 2) \) and \( (2, -3) \) can be calculated using the formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Where \( (x_1, y_1) = (-3, 2) \) and \( (x_2, y_2) = (2, -3) \).

Let's calculate this:
\[
M_x = \frac{-3 + 2}{2} = \frac{-1}{2}
\]
\[
M_y = \frac{2 + (-3)}{2} = \frac{-1}{2}
\]
So, the midpoint \( M \) is \( \left(-\frac{1}{2}, -\frac{1}{2}\right) \).

2. **Find the Slope of the Line Between \( Y \) and \( Y' \)**:
Next, we find the slope of the line segment connecting \( Y \) and \( Y' \):
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{2 - (-3)} = \frac{-5}{5} = -1
\]

3. **Find the Slope of the Line of Reflection**:
The line of reflection is perpendicular to the line connecting the original and reflected points, so its slope will be the negative reciprocal of -1, which is 1.

4. **Equation of the Line of Reflection**:
Using the slope and the midpoint, we can now use the point-slope form of the equation of a line:
\[
y - y_1 = m (x - x_1)
\]
Here \( m = 1 \) and \( (x_1, y_1) = \left(-\frac{1}{2}, -\frac{1}{2}\right) \).

Plugging in the values:
\[
y - \left(-\frac{1}{2}\right) = 1 \left(x - \left(-\frac{1}{2}\right)\right)
\]
Simplifying this:
\[
y + \frac{1}{2} = x + \frac{1}{2}
\]
\[
y = x
\]

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