Question

math: rigid motions (moves to terms)
segments ( overline{abc} ) and ( overline{def} ) are shown in the figure to the right. ( \triangle abc ) is mapped to ( \triangle def ) using a translation 14 units and then a reflection across the line ( y = x ). which of the following explains the relationship between the corresponding lengths of ( overline{abc} ) and ( overline{def} )?

a) since the corresponding lengths in a rigid transformation (such as a translation or reflection) are congruent, the length of ( overline{abc} ) is equal to the length of ( overline{def} ).

b) since lines ( overline{ab} ) and ( overline{de} ) are parallel (due to the translation), the image of one segment will coincide with the other after a rigid transformation (a reflection). therefore, the length of ( overline{abc} ) is equal to the length of ( overline{def} ).

c) since the length of ( overline{ab} ) is three times the length of ( overline{de} ), this is a non - rigid transformation (a dilation), so the length of ( overline{abc} ) is three times the length of ( overline{def} ).

d) since the corresponding lengths of two similar figures (with different shapes) are in a ratio after a non - rigid transformation (such as a dilation), the length of ( overline{abc} ) is proportional to the length of ( overline{def} ) with the ratio equal to the scale factor of the dilation.

Explanation:

To determine the correct reason for the congruence of \( \triangle DEF \) and \( \triangle D'E'F' \) after a translation, we analyze each option:

• Option A: Rotation involves turning around a point, but the problem is about translation (sliding), so this is incorrect.
• Option B: Translation is a rigid transformation that slides a figure without rotating or reflecting. It preserves side lengths and angle measures, so corresponding sides \( AB \) and \( DE \) (wait, in the problem it's \( \triangle DEF \) and \( \triangle D'E'F' \)) – during translation, the image is congruent to the pre - image because translation is a rigid motion. This option correctly states that translation is a rigid transformation that maps the pre - image to the image with preserved side lengths and angle measures, so the triangles are congruent.
• Option C: Dilation changes the size of the figure (it's a non - rigid transformation), but translation doesn't involve dilation, so this is incorrect.
• Option D: Reflection is a flip over a line, while the problem is about translation (slide), so this is incorrect.

Answer:

B. Since sides \( AB \) and \( DE \) are corresponding sides, the image of one triangle will coincide with the other triangle by a rigid transformation (a translation). Therefore, \( \triangle DEF\cong\triangle D'E'F' \)