To determine the line of reflection that produces [tex]\( X' (2, 4) \)[/tex] from [tex]\( X (-2, 4) \)[/tex] in triangle [tex]\( XYZ \)[/tex], we need to understand how reflections work. Specifically, reflections preserve the y-coordinates when the reflection is across a vertical line, and they change the x-coordinates symmetrically about the line of reflection.
1. First, let’s observe the original and reflected coordinates for point [tex]\( X \)[/tex]:
- Original [tex]\( X \)[/tex] is at [tex]\( (-2, 4) \)[/tex]
- Reflected [tex]\( X' \)[/tex] is at [tex]\( (2, 4) \)[/tex]
2. The y-coordinates remain the same ([tex]\( y = 4 \)[/tex]), suggesting that the reflection line must be vertical.
3. To find the exact vertical line of reflection, we need to determine the midpoint between [tex]\( X \)[/tex] and [tex]\( X' \)[/tex]. The midpoint formula for two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
4. Applying the midpoint formula to our points [tex]\( X \)[/tex] and [tex]\( X' \)[/tex]:
[tex]\[
\left( \frac{-2 + 2}{2}, \frac{4 + 4}{2} \right) = \left( \frac{0}{2}, \frac{8}{2} \right) = (0, 4)
\][/tex]
5. The midpoint [tex]\( (0, 4) \)[/tex] lies exactly on the y-axis, indicating that the line of reflection is where the x-coordinate equals zero.
6. Therefore, the line of reflection that produces [tex]\( X' (2, 4) \)[/tex] from [tex]\( X (-2, 4) \)[/tex] is the y-axis.
So, the correct answer is:
[tex]\[ \text{the y-axis} \][/tex]
