Question

triangle xyz is transformed to create triangle xyz. the side lengths of both triangles are shown.
\\( \overline { xy } = 10 \mathrm { cm } \quad \overline { x ^ { \prime } y ^ { \prime } } = 10 \mathrm { cm } \\)
\\( \overline { y z } = 9 \mathrm { cm } \quad \overline { y ^ { \prime } z ^ { \prime } } = 9 \mathrm { cm } \\)
\\( \overline { z x } = 4 \mathrm { cm } \quad \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 (also called an isometry) is a transformation that preserves the side - lengths and angles of the pre - image. In this case, we can see that \(\overline{XY}=\overline{X'Y'}=10\mathrm{cm}\), \(\overline{YZ}=\overline{Y'Z'}=9\mathrm{cm}\) and \(\overline{ZX}=\overline{Z'X'}=4\mathrm{cm}\). Since the pre - image \(XYZ\) and the image \(X'Y'Z'\) have the same side - length measures, the transformation is rigid.

Option 2 is incorrect because not all transformations are rigid (e.g., dilations are non - rigid). Option 3 is wrong as the side lengths of \(\triangle XYZ\) are not different from each other in the sense of non - preservation (they are preserved). Option 4 is incorrect because there is no indication of a scaling factor of \(\frac{1}{2}\) (in fact, the side lengths are equal).

Answer:

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