Question

point f on the coordinate grid is reflected across a line to create point f. then, points e, f, and f are connected to create a triangle. karl says point e must be on the perpendicular bisector of the line ff, regardless of the line across which point f was reflected.

Explanation:

Step1: Recall reflection property

When a point is reflected across a line, the line of reflection is the perpendicular bisector of the segment connecting the point and its image. Let the line of reflection be \(l\). So, \(l\) is the perpendicular bisector of \(FF'\).

Step2: Consider the condition for point \(E\)

For point \(E\) to be on the perpendicular bisector of \(FF'\), we can choose a line of reflection such that the perpendicular - bisector of \(FF'\) passes through \(E\). If we reflect point \(F\) across the line \(x = 0\) (the \(y\) - axis).
Let \(F=(5,3)\). When \(F\) is reflected across the \(y\) - axis (\(x = 0\)), the image \(F'=(- 5,3)\). The mid - point of the line segment \(FF'\) is \((\frac{5+( - 5)}{2},\frac{3 + 3}{2})=(0,3)\). The line \(x = 0\) is the perpendicular bisector of \(FF'\), and point \(E=(0,1)\) lies on the \(y\) - axis (\(x = 0\)).

Answer:

\(x = 0\)