Question

a sequence of transformations is described below.

• a dilation about a point ( p )
• a rotation about another point ( q )
• a reflection over the line ( overleftrightarrow{pq} )

which of the following must be preserved under this sequence of transformations?
choose 1 answer:
a angle measures only
b segment lengths only
c both angle measures and segment lengths
d neither angle measures nor segment lengths

Explanation:

Step1: Analyze Dilation

Dilation is a transformation that scales the figure (changes segment lengths) but preserves angle measures. So after dilation, angle measures are preserved, segment lengths are not (unless scale factor is 1, but dilation in general changes lengths).

Step2: Analyze Rotation

Rotation is a rigid transformation. Rigid transformations (rotation, reflection, translation) preserve both angle measures and segment lengths.

Step3: Analyze Reflection

Reflection is also a rigid transformation, so it preserves both angle measures and segment lengths.

Step4: Combine Transformations

The sequence has dilation (preserves angles, changes lengths) followed by rotation (rigid, preserves both) and reflection (rigid, preserves both). But the key is: dilation preserves angle measures, and rotation/reflection preserve both. However, the question is what MUST be preserved. Dilation preserves angles, and rotation/reflection preserve angles and lengths. But since dilation is part of the sequence, segment lengths are not preserved (because dilation can change them), but angle measures are preserved through all transformations (dilation preserves angles, rotation and reflection also preserve angles). Wait, no—wait: Dilation preserves angle measures (angles stay the same), but changes segment lengths. Rotation and reflection are rigid, so they preserve both. But the sequence is dilation first, then rotation, then reflection. So after dilation, angles are preserved, lengths are scaled. Then rotation: rigid, so angles and lengths (but lengths were already scaled by dilation, so rotation preserves the scaled lengths). Then reflection: rigid, preserves angles and scaled lengths. Wait, no—actually, the question is what is preserved under the entire sequence. Let's recall: Dilation is a similarity transformation, which preserves angle measures (angles are congruent) but not necessarily segment lengths (unless scale factor 1). Rotation and reflection are congruence transformations (rigid), which preserve both angle measures and segment lengths. So in the sequence: dilation (similarity) → rotation (congruence) → reflection (congruence). The dilation preserves angle measures, and the subsequent rigid transformations also preserve angle measures. For segment lengths: dilation can change them (unless scale factor 1, but the problem doesn't say scale factor 1), so segment lengths are not necessarily preserved (because dilation can scale them). But angle measures: dilation preserves them, and rotation/reflection also preserve them. So angle measures are preserved, segment lengths are not (because dilation can change them). Wait, but wait—maybe I made a mistake. Let's check the options. Option A: Angle measures only. Let's verify each transformation:

- Dilation: preserves angle measures, changes segment lengths.
- Rotation: preserves both angle measures and segment lengths (but segment lengths were already changed by dilation, so rotation preserves the new lengths, but the original lengths are not preserved—wait, no, the question is about what is preserved under the sequence. So after dilation, the figure has scaled lengths. Then rotation: preserves the scaled lengths (so lengths are preserved relative to the dilated figure, but not relative to the original). But the question is: what is preserved (i.e., remains the same) from the original figure to the final image. So original figure → dilated (angles same, lengths scaled) → rotated (angles same, lengths same as dilated) → reflected (angles same, lengths same as dilated). So from original to final: angles are same (because dilation preserves angles, rotation/reflection preserve angles), lengths are scaled (because dilation scales them, and rotation/reflection preserve the scaled lengths). Therefore, angle measures are preserved, segment lengths are not (unless scale factor 1, but the problem doesn't specify that). Therefore, only angle measures are preserved. Wait, but let's check the options again. Option A: Angle measures only. Option C: Both. Wait, maybe I messed up. Wait, no—dilation is a similarity transformation, which means that angle measures are preserved (angles are congruent), and segment lengths are proportional (not necessarily equal). Then rotation and reflection are congruence transformations, which preserve both angle measures and segment lengths (make them congruent). But in the sequence, the first transformation is dilation (similarity), so the image after dilation is similar to the original (angles congruent, sides proportional). Then rotation: congruence, so the image after rotation is congruent to the dilated image (so angles congruent, sides congruent to dilated image). Then reflection: congruence, so image after reflection is congruent to the rotated image (so angles congruent, sides congruent to rotated image). Therefore, from original to final image: angles are congruent (preserved), sides are proportional (from dilation) and then congruent to the dilated sides (so sides are scaled by the dilation factor, then preserved in length relative to the dilated image). But the question is: what is preserved (i.e., remains the same) between the original and the final image. So angle measures: same (preserved). Segment lengths: not same (scaled by dilation factor, unless factor 1). Therefore, angle measures are preserved, segment lengths are not. Therefore, the answer is A: Angle measures only. Wait, but let's check the options again. Option A: Angle measures only. Option C: Both. Wait, maybe I made a mistake. Wait, no—dilation preserves angle measures, rotation and reflection preserve angle measures and segment lengths. But the sequence includes dilation, which changes segment lengths. So segment lengths are not preserved (because dilation can change them), but angle measures are preserved through all transformations. So the correct answer is A.

Answer:

A. Angle measures only