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Which sequence of transformations would result in an image that is similar and non-congruent to the pre-image? Assume that any scale factor for the dilations is not equal to 1. Check all that apply.

a dilation followed by a translation
a dilation followed by a reflection
a reflection followed by a dilation
a rotation followed by a reflection
a rotation followed by a dilation
a translation followed by a reflection

1 answer

To determine which transformations result in an image that is similar (same shape, possibly different size) and non-congruent (not the same size) to the pre-image, we need to identify transformations that incorporate a dilation (scaling) factor not equal to 1 along with other transformations.

A dilation by itself changes the size of the image unless the scale factor is 1, making it non-congruent to the original. When combined with another transformation (like translation, reflection, or rotation), it should still maintain the property of similarity but alter the congruency.

Let's analyze the options:

  1. A dilation followed by a translation: This results in a similar image, since the dilation changes the size, and translation does not affect size. (Similar and non-congruent)

  2. A dilation followed by a reflection: Again, the dilation changes the size, and reflection remains a rigid transformation. This will produce a similar shape that is also non-congruent. (Similar and non-congruent)

  3. A reflection followed by a dilation: The reflection does not change the size, but the subsequent dilation does. Hence, this results in a similar and non-congruent image. (Similar and non-congruent)

  4. A rotation followed by a reflection: Both rotation and reflection are rigid transformations that do not change size. The image produced here is congruent, not just similar. (Not similar and non-congruent)

  5. A rotation followed by a dilation: The rotation does not change size, but dilation does. This results in a similar and non-congruent image. (Similar and non-congruent)

  6. A translation followed by a reflection: Both of these are rigid transformations and do not affect the size of the image. Thus, this sequence will result in a congruent image. (Not similar and non-congruent)

Based on the analysis, the transformations that yield an image that is similar and non-congruent to the pre-image are:

  • A dilation followed by a translation
  • A dilation followed by a reflection
  • A reflection followed by a dilation
  • A rotation followed by a dilation

So, the correct transformations are:

  • A dilation followed by a translation
  • A dilation followed by a reflection
  • A reflection followed by a dilation
  • A rotation followed by a dilation