Answer :
To determine which composition of transformations will create a pair of similar, not congruent triangles, we need to understand the effects of each type of transformation.
1. Rotation: Rotating a figure around a point results in another figure that is congruent to the original. The size and shape remain unchanged, only the orientation is different.
2. Translation: Translating a figure involves sliding it in any direction without changing its orientation or size. The resulting figure is congruent to the original.
3. Reflection: Reflecting a figure over a line creates a mirror image. The size and shape remain the same, so the resulting figure is congruent to the original.
4. Dilation: Dilation transforms a figure by resizing it. The shape of the figure remains the same but the size changes, which creates similar figures (figures with the same shape but different sizes), not congruent ones.
Considering the options provided:
- Option 1: "A rotation, then a reflection." Both rotation and reflection individually result in congruent figures since they only change orientation but not the size. The combination of these two transformations also results in a congruent figure, hence they do not produce similar but not congruent triangles.
- Option 2: "A translation, then a rotation." Both translation and rotation preserve the size and shape of a figure. Hence, their combination results in another congruent triangle, not similar.
- Option 3: "A reflection, then a translation." Similar to the above reasoning, a reflection preserves the size and shape and so does translation. Their combination produces a congruent triangle.
- Option 4: "A rotation, then a dilation." Rotation will preserve the shape and size initially, but dilation will change the size while keeping the shape the same. Hence, one rotation followed by a dilation will result in triangles that are similar (same shape) but not congruent (different sizes).
Therefore, the correct composition that will create a pair of similar, not congruent triangles is:
- A rotation, then a dilation
Hence, the answer is option 4.
1. Rotation: Rotating a figure around a point results in another figure that is congruent to the original. The size and shape remain unchanged, only the orientation is different.
2. Translation: Translating a figure involves sliding it in any direction without changing its orientation or size. The resulting figure is congruent to the original.
3. Reflection: Reflecting a figure over a line creates a mirror image. The size and shape remain the same, so the resulting figure is congruent to the original.
4. Dilation: Dilation transforms a figure by resizing it. The shape of the figure remains the same but the size changes, which creates similar figures (figures with the same shape but different sizes), not congruent ones.
Considering the options provided:
- Option 1: "A rotation, then a reflection." Both rotation and reflection individually result in congruent figures since they only change orientation but not the size. The combination of these two transformations also results in a congruent figure, hence they do not produce similar but not congruent triangles.
- Option 2: "A translation, then a rotation." Both translation and rotation preserve the size and shape of a figure. Hence, their combination results in another congruent triangle, not similar.
- Option 3: "A reflection, then a translation." Similar to the above reasoning, a reflection preserves the size and shape and so does translation. Their combination produces a congruent triangle.
- Option 4: "A rotation, then a dilation." Rotation will preserve the shape and size initially, but dilation will change the size while keeping the shape the same. Hence, one rotation followed by a dilation will result in triangles that are similar (same shape) but not congruent (different sizes).
Therefore, the correct composition that will create a pair of similar, not congruent triangles is:
- A rotation, then a dilation
Hence, the answer is option 4.
