Question

Triangle XYZ is drawn with vertices X(4, −5), Y(6, −1), Z(10, −8). Determine the line of reflection if Y′(−6, −1).

y-axis
x-axis
y = −1
x = 6

Answer 1

To find the line of reflection given a point \( Y(6, -1) \) and its reflection \( Y'(-6, -1) \), we start by determining the midpoint of the segment connecting \( Y \) and \( Y' \).

The midpoint \( M \) of the segment connecting the points \( (6, -1) \) and \( (-6, -1) \) is given by:

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

Substituting the coordinates of \( Y \) and \( Y' \):

\[
M = \left( \frac{6 + (-6)}{2}, \frac{-1 + (-1)}{2} \right) = \left( \frac{0}{2}, \frac{-2}{2} \right) = (0, -1)
\]

Next, we need to determine the line of reflection, which is the vertical line halfway between the points on the x-axis. The x-coordinates of \( Y \) and \( Y' \) are 6 and -6, respectively. The line of reflection lies on the line that is perpendicular to the segment connecting \( Y \) and \( Y' \) and goes through the midpoint \( M(0, -1) \).

Since \( Y \) and \( Y' \) both have the same y-coordinate (-1), the line of reflection must be vertical. The x-coordinate of the midpoint (which is 0) indicates that the line of reflection is the y-axis.

Thus, the correct line of reflection is:

\[
\text{y-axis}
\]

The correct answer is:
**y-axis**