A transformation was applied to $X Y Z$ to create $X^{\prime} Y^{\prime} Z^{\prime}$. Which statement explains whether or not the transformation is isometric?
A. It is isometric because both figures have equal corresponding side lengths and angles.
B. It is isometric because both figures are right triangles.
C. It is not isometric because corresponding angle measures are the same but corresponding side lengths are different.
D. It is not isometric because there was no change in shape.
Answer (2)
An isometric transformation preserves distances, meaning corresponding side lengths and angles are equal.
The statement "It is isometric because both figures have equal corresponding side lengths and angles" accurately describes an isometric transformation.
Other options are incorrect because they either don't guarantee isometry (e.g., being right triangles) or misrepresent non-isometric transformations.
Therefore, the correct explanation is that the figures have equal corresponding side lengths and angles.
Explanation
Understanding Isometric Transformations An isometric transformation is a transformation that preserves distances between points. This means that the corresponding side lengths of the original and transformed figures must be equal. Also, corresponding angles must be equal.
Analyzing the First Statement The first statement says: "It is isometric because both figures have equal corresponding side lengths and angles." This statement correctly describes an isometric transformation.
Analyzing the Second Statement The second statement says: "It is isometric because both figures are right triangles." This is incorrect. While both figures could be right triangles, this alone does not guarantee that the transformation is isometric. The side lengths must also be equal.
Analyzing the Third Statement The third statement says: "It is not isometric because corresponding angle measures are the same but corresponding side lengths are different." This statement correctly describes a non-isometric transformation. If the side lengths are different, the distances between points are not preserved, so the transformation is not isometric.
Analyzing the Fourth Statement The fourth statement says: "It is not isometric because there was no change in shape." This is incorrect. If there is no change in shape and size, the transformation is isometric (it's the identity transformation).
Determining the Correct Statement The first and third statements are correct. However, the question asks for which statement explains whether or not the transformation is isometric. The first statement says the transformation is isometric, while the third statement says the transformation is not isometric. Since we don't know if the side lengths are the same or different, we cannot definitively say whether the transformation is isometric or not. However, the question is asking which statement explains whether or not the transformation is isometric. The first statement explains what it means for a transformation to be isometric, and the third statement explains what it means for a transformation to not be isometric. The question is asking for a single statement. The first statement is the most accurate and complete explanation of an isometric transformation.
Final Answer The correct statement is: It is isometric because both figures have equal corresponding side lengths and angles.
Examples
Imagine you're creating a logo for a company. If you need to resize the logo without distorting its proportions, you're essentially applying an isometric transformation. This ensures that the logo remains recognizable and consistent, regardless of its size. Similarly, in architecture, scaling blueprints while maintaining the building's proportions is an isometric transformation, preserving the design's integrity.
The transformation from triangle X Y Z to triangle X ′ Y ′ Z ′ is isometric if both figures have equal corresponding side lengths and angles. Hence, the correct response is Statement A: 'It is isometric because both figures have equal corresponding side lengths and angles.' Understanding isometric transformations is important for proving figures are congruent in geometry.
;
