Question

Triangle XYZ is drawn with vertices X(−2, 4), Y(−9, 3), Z(−10, 7). Determine the line of reflection that produces Z′(10, 7).

y-axis
x-axis
y = 3
x = −4

Answer 1

To determine the line of reflection that produces the point \( Z'(10, 7) \) from the point \( Z(-10, 7) \), we will first find the midpoint of the line segment connecting \( Z \) and \( Z' \).

The coordinates of \( Z \) are \( (-10, 7) \) and the coordinates of \( Z' \) are \( (10, 7) \).

The midpoint \( M \) of the segment \( ZZ' \) can be calculated using the midpoint formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting in the coordinates of \( Z \) and \( Z' \):
\[
M = \left( \frac{-10 + 10}{2}, \frac{7 + 7}{2} \right) = \left( \frac{0}{2}, \frac{14}{2} \right) = (0, 7)
\]

Next, we determine the line of reflection. The line of reflection is the perpendicular bisector of the segment connecting \( Z \) and \( Z' \). Since both points \( Z \) and \( Z' \) have the same y-coordinate (7), the line connecting these points is horizontal. Therefore, the line of reflection must be vertical, since it needs to be perpendicular to the segment.

The reflection must also go through the midpoint \( (0, 7) \). Thus, the line of reflection is the vertical line that has the equation:
\[
x = 0
\]
This line is the y-axis.

Given the options:
- y-axis
- x-axis
- y = 3
- x = −4

The correct answer is:
\[
\text{y-axis}
\]