- Recognize that rotations, translations, and reflections are rigid motions that preserve size and shape, resulting in congruent triangles.
- Identify that a dilation is a similarity transformation that changes the size of a figure (for scale factor \(k
eq 1\)), resulting in figures that are similar but not congruent.
- Conclude that the composition involving a dilation—specifically a rotation then a dilation—produces triangles that are similar but not congruent.
- The correct choice is \(\boxed{\text{a rotation, then a dilation}}\).
Explanation
- Analyze Rigid Motions and Congruence
To determine which composition of transformations creates similar but not congruent triangles, we first need to define the fundamental types of geometric transformations.
A rigid motion, or isometry, is a transformation that preserves the size and shape of a figure. This means that if two figures are related by a rigid motion, they are congruent. The most common rigid motions are:
- Translation: Sliding a figure in the plane.
- Rotation: Turning a figure around a fixed point.
- Reflection: Flipping a figure over a line.
Since these transformations preserve both side lengths and angle measures, they result in congruent triangles. Any composition of these three operations—whether it be rotation, reflection, or translation—remains a rigid motion, meaning the resulting triangles will always be congruent.
- Analyze Dilations and Similarity Transformations
A dilation is a different type of transformation known as a similarity transformation. A dilation scales a figure by a factor of \(k\) from a center point.
- If \(k = 1\), the figure remains the same size (congruent).
- If \(k
eq 1\), the figure's size changes.
Because dilations preserve angle measures, they ensure that the resulting figures remain similar. However, because they alter the side lengths (provided \(k
eq 1\)), the figures are no longer congruent. Therefore, any composition that includes a dilation where \(k
eq 1\) will produce figures that are similar but not congruent.
- Evaluate the Options and Conclude
Now, let's examine the options:
- Rotation, then reflection: Both are rigid motions. The composition is a rigid motion, so the triangles are congruent.
- Translation, then rotation: Both are rigid motions. The composition is a rigid motion, so the triangles are congruent.
- Reflection, then translation: Both are rigid motions. The composition is a rigid motion, so the triangles are congruent.
- Rotation, then dilation: This composition includes a dilation. Since a dilation changes the size of the figure (for \(k
eq 1\)), the resulting triangle will be similar to the original but not congruent.
Thus, the composition of a rotation followed by a dilation is the correct choice.
Examples
Consider a digital image editor that allows you to manipulate photos. When you rotate a photo or flip it (reflection), you are performing rigid motions; the content of the image retains its original dimensions and proportions—the objects in the photo are congruent to the original. However, when you use a "zoom" or "resize" feature, you are applying a dilation. If you rotate a picture and then resize it, the objects in the final image are similar in shape to the original objects, but they have a different scale, meaning they are not congruent. This helps in understanding how geometric transformations are applied in everyday software applications.