Question

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

Explanation:

Step1: Recall rigid transformation definition

A rigid transformation (isometry) preserves the side lengths and angle measures of a figure. So we check if the corresponding sides of the pre - image (triangle XYZ) and image (triangle X'Y'Z') have the same lengths.

Step2: Compare corresponding side lengths

We are given:

• For side \( \overline{XY} \) and \( \overline{X'Y'} \): \( \overline{XY}=10\space\text{cm} \) and \( \overline{X'Y'}=10\space\text{cm} \)
• For side \( \overline{YZ} \) and \( \overline{Y'Z'} \): \( \overline{YZ} = 9\space\text{cm}\) and \( \overline{Y'Z'}=9\space\text{cm} \)
• For side \( \overline{ZX} \) and \( \overline{Z'X'} \): \( \overline{ZX}=4\space\text{cm} \) and \( \overline{Z'X'}=4\space\text{cm} \)

All corresponding sides have the same length. Now let's analyze the options:

• Option 1: "Yes, the pre - image and image have the same side length measures." This matches our analysis since all corresponding sides are equal, so it is a rigid transformation.
• Option 2: "Yes, all transformations are rigid." This is false. For example, dilation is a non - rigid transformation as it changes side lengths.
• Option 3: "No, the side lengths of triangle XYZ are different from each other." The fact that the sides of the pre - image are different from each other does not affect whether the transformation is rigid. A rigid transformation just needs to preserve the side lengths (so corresponding sides are equal), not that the pre - image sides are equal to each other.
• Option 4: "No, each side length of X'Y'Z' should be one half of the corresponding side length of XYZ." There is no reason for the side lengths of the image to be half of the pre - image. The given side lengths show that the corresponding sides are equal, not scaled by a factor of \( \frac{1}{2} \).

Answer:

Yes, the pre - image and image have the same side length measures. (The correct option is the first one: "Yes, the pre - image and image have the same side length measures.")