Which composition of transformations will create a pair of similar, not congruent triangles?
A. a rotation, then a reflection
B. a translation, then a rotation
C. a reflection, then a translation
D. a rotation, then a dilation
Answer (2)
The composition of transformations that creates similar, not congruent triangles is a rotation followed by a dilation. Rigid transformations like rotations, reflections, and translations preserve size and shape, resulting in congruent figures. Therefore, option D is the correct choice.
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Rotations, reflections, and translations are rigid transformations that preserve size and shape, resulting in congruent figures.
Dilation is a non-rigid transformation that changes the size but preserves the shape, resulting in similar figures.
A composition including a dilation will result in similar, but not congruent figures.
The composition of a rotation, then a dilation will create similar, not congruent triangles. A ro t a t i o n , t h e na d i l a t i o n
Explanation
Understanding Transformations To determine which composition of transformations will create similar, but not congruent triangles, we need to understand the properties of each transformation.
Rigid Transformations Rotations, reflections, and translations are rigid transformations. Rigid transformations preserve the size and shape of the figure. Therefore, if we only use these transformations, the resulting triangles will be congruent.
Non-Rigid Transformations Dilation is a non-rigid transformation. Dilation changes the size of the figure but preserves its shape. Therefore, if we include a dilation in the composition, the resulting triangles will be similar, but not congruent.
Analyzing the Options Now, let's examine the options:
A rotation, then a reflection: Both are rigid transformations, so the triangles will be congruent.
A translation, then a rotation: Both are rigid transformations, so the triangles will be congruent.
A reflection, then a translation: Both are rigid transformations, so the triangles will be congruent.
A rotation, then a dilation: A dilation is a non-rigid transformation, so the triangles will be similar, but not congruent.
Final Answer Therefore, the composition of transformations that will create a pair of similar, not congruent triangles is a rotation, then a dilation.
Examples
Imagine you have a photograph. If you rotate, reflect, or translate the photo, it remains the same size and shape. However, if you enlarge (dilate) the photo, the size changes, but the proportions remain the same, creating a similar, but not congruent, image. This concept applies to various fields, such as art, architecture, and engineering, where scaling objects while maintaining their proportions is essential.
