To determine the line of reflection that produces [tex]\( Z'(10, 7) \)[/tex] from [tex]\( Z(-10, 7) \)[/tex], we need to understand the properties of reflections in geometry.
A reflection flips a point or figure over a line in such a way that the original point and its image are equidistant from the line of reflection. Here’s the step-by-step process:
1. Identify Coordinates of the Point and its Image:
- Original point [tex]\( Z \)[/tex] has coordinates [tex]\((-10, 7)\)[/tex].
- Reflected image [tex]\( Z' \)[/tex] has coordinates [tex]\( (10, 7) \)[/tex].
2. Determine the Midpoint:
- The line of reflection lies midway between the original point [tex]\( Z \)[/tex] and its reflected image [tex]\( Z' \)[/tex].
- To find the midpoint, we calculate:
[tex]\[
\text{midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Using the coordinates:
[tex]\[
\text{midpoint} = \left( \frac{-10 + 10}{2}, \frac{7 + 7}{2} \right) = (0, 7)
\][/tex]
3. Determine the Line of Reflection:
- The y-coordinates of both points are the same (7), which means the line of reflection is vertical and must pass through the x-coordinate of the midpoint.
- Since the x-coordinate of the midpoint is [tex]\( 0 \)[/tex], the line is:
[tex]\[
x = 0
\][/tex]
In conclusion, the line of reflection that produces [tex]\( Z'(10, 7) \)[/tex] from [tex]\( Z(-10, 7) \)[/tex] is the y-axis. This can be described mathematically as [tex]\( x = 0 \)[/tex].
