Question

1. here are triangles eki and eti. reflect triangle eki across line ei. without any additional justification, which of these is a valid reason why the image of k will coincide with t? a. the image of k and t are the same distance along the same ray from e. b. line ei is the perpendicular bisector of the segment connecting k and t because the perpendicular bisector is determined by 2 points that are both equidistant from the endpoints of a segment. c. the image of k and t coincide after reflection because we defined our transformation that way. d. the image of k and t are on the same side of line ei and make the same angle with it at e.

Explanation:

• Option A: Reflection across a line does not imply points are on the same ray. So this is incorrect.
• Option B: If a line is the perpendicular bisector of a segment, then the two endpoints of the segment are symmetric with respect to that line. In a reflection across line \(EI\), if \(EI\) is the perpendicular bisector of \(KT\), then the image of \(K\) (after reflection across \(EI\)) will coincide with \(T\).
• Option C: Just defining a transformation does not give a valid geometric reason. This is a circular - like argument without geometric basis.
• Option D: In a reflection, the pre - image and image are on opposite sides of the line of reflection. So this is incorrect.

Answer:

B. Line \(EI\) is the perpendicular bisector of the segment connecting \(K\) and \(T\) because the perpendicular bisector is determined by 2 points that are both equidistant from the endpoints of a segment.