Question

triangle xyz is transformed to create triangle xyz. the side lengths of both triangles are shown.
\\( \overline { xy } = 10 \mathrm { cm } \\) \\( \overline { x ^ { \prime } y ^ { \prime } } = 10 \mathrm { cm } \\)
\\( \overline { y z } = 9 \mathrm { cm } \\) \\( \overline { y ^ { \prime } z ^ { \prime } } = 9 \mathrm { cm } \\)
\\( \overline { z x } = 4 \mathrm { cm } \\) \\( \overline { z ^ { \prime } x ^ { \prime } } = 4 \mathrm { cm } \\)
is this a rigid transformation? explain.
yes, the pre - image and image have the same side length measures.
yes, all transformations are rigid.
no, the side lengths of triangle xyz are different from each other.
no, each side length of xyz should be one half of the corresponding side length of xyz.

Explanation:

A rigid transformation preserves the side lengths and angle measures of a figure. Here, the side lengths of triangle \( XYZ \) and its image \( X'Y'Z' \) are equal (\( \overline{XY}=\overline{X'Y'}=10 \) cm, \( \overline{YZ}=\overline{Y'Z'}=9 \) cm, \( \overline{ZX}=\overline{Z'X'}=4 \) cm). The second option is wrong because not all transformations are rigid (e.g., dilations are not). The third option is wrong as the side lengths of \( XYZ \) being different from each other doesn't affect if it's a rigid transformation (a triangle can have different side lengths). The fourth option is wrong as there's no indication of scaling. So the correct reasoning is that the pre - image and image have the same side length measures.

Answer:

A. Yes, the pre - image and image have the same side length measures.