Question

triangle xyz is transformed to create triangle xyz. the side lengths of both triangles are shown.
\\( \overline { x y } = 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.
o yes, the pre - image and image have the same side length measures.
o yes, all transformations are rigid.
o no, the side lengths of triangle xyz are different from each other.
o no, each side length of xyz should be one half of the corresponding side length of xyz.

Explanation:

A rigid transformation (such as translation, rotation, reflection) preserves the side - lengths of the figure. Here, we check the side - lengths of the pre - image (\(\triangle XYZ\)) and the image (\(\triangle X'Y'Z'\)).

For \(\overline{XY}\) and \(\overline{X'Y'}\): \(\overline{XY} = 10\mathrm{cm}\) and \(\overline{X'Y'}=10\mathrm{cm}\)

For \(\overline{YZ}\) and \(\overline{Y'Z'}\): \(\overline{YZ} = 9\mathrm{cm}\) and \(\overline{Y'Z'}=9\mathrm{cm}\)

For \(\overline{ZX}\) and \(\overline{Z'X'}\): \(\overline{ZX} = 4\mathrm{cm}\) and \(\overline{Z'X'}=4\mathrm{cm}\)

Since the side - lengths of the pre - image and the image are equal, the transformation is rigid.

Option 2 is wrong because not all transformations are rigid (e.g., dilation is non - rigid). Option 3 is wrong as side - lengths of \(\triangle XYZ\) are not compared to itself but to \(\triangle X'Y'Z'\). Option 4 is wrong as there is no indication of a scaling factor of \(\frac{1}{2}\) (side - lengths are equal).

Answer:

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