ΔA'B'C' was constructed using ΔABC and line segment EH.

Question

2 triangles are shown. Line E H is the line of reflection. Line segment B B prime has a midpoint at point D. Line segment A A prime has a midpoint at point F. Line segment C C prime has a midpoint at point G.

For to be the line of reflection between and , which statements must be true? Select three options.

BD = DB'
DF = FG
m∠EFA = 90°
The line of reflection, EH, is the perpendicular bisector of BB', AA', and CC'.
ΔABC is not congruent to ΔA'B'C'.

Answer 1

To determine the necessary conditions for line segment EH to be the line of reflection between triangles ΔABC and ΔA'B'C', the following statements must be true:

1. The line of reflection, EH, is the perpendicular bisector of BB', AA', and CC'.
   This statement is true and is a fundamental property of reflections: the line of reflection bisects each segment connecting corresponding points of the original triangle and its image, and does so perpendicularly.

2. BD = DB'
   This is true because, for EH to be the line of reflection, point D must be the midpoint of segment BB', indicating that the distances from D to B and D to B' are equal.

3. m∠EFA = 90°
   This statement is also true. For EH to serve as the line of reflection, it must be perpendicular to segment AA'. Since F is the midpoint of AA', the angle formed between the line of reflection and the segment must be 90 degrees.

Combining these, the three statements you should select are:

1. The line of reflection, EH, is the perpendicular bisector of BB', AA', and CC'.
2. BD = DB'.
3. m∠EFA = 90°.

The statements regarding DF = FG and ΔABC is not congruent to ΔA'B'C' are not necessary for EH to be the line of reflection. In fact, any two triangles that are reflections of each other across a line are congruent. Thus, ΔABC would indeed be congruent to ΔA'B'C'.