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 y}=10 \mathrm{~cm} \\)
\\( \overline{y z}=9 \mathrm{~cm} \overline{y z}=9 \mathrm{~cm} \\)
\\( \overline{z x}=4 \mathrm{~cm} \overline{z x}=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 (also called an isometry) is a transformation that preserves the side - lengths and angles of a figure. In this case, since \(\overline{XY}=\overline{X'Y'}=10\mathrm{cm}\), \(\overline{YZ}=\overline{Y'Z'}=9\mathrm{cm}\), and \(\overline{ZX}=\overline{Z'X'}=4\mathrm{cm}\), the pre - image (\(\triangle XYZ\)) and the image (\(\triangle X'Y'Z'\)) have the same side - length measures.
The statement “Yes, all transformations are rigid” is incorrect because non - rigid transformations (such as dilations) exist. The statement “No, the side lengths of triangle \(XYZ\) are different from each other” is not relevant to determining if the transformation is rigid (rigid vs non - rigid is about the relationship between pre - image and image, not about the internal side - length relationships of the pre - image). The statement “No, each side length of \(X'Y'Z'\) should be one half of the corresponding side length of \(XYZ\)” is wrong as the side lengths are equal, not in a \(1:2\) ratio.

Answer:

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