Question

a sequence of transformations is described below.

• a dilation about a point \\(p\\)
• a rotation about \\(p\\)
• a horizontal stretch about a vertical line \\(\overleftrightarrow{qr}\\)

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:

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<reasoning>

Analyze the first two transformations

Using the Geometric Transformations and Rigid Transformations knowledge points

• A dilation about a point \(P\) preserves angle measures but changes segment lengths.
• A rotation about \(P\) is a rigid transformation, preserving both angle measures and segment lengths.

Analyze the horizontal stretch

• A horizontal stretch about a vertical line \(\overleftrightarrow{QR}\) scales coordinates in only one direction.
• This non-rigid transformation distorts shapes, changing both segment lengths and angle measures.

Determine what is preserved overall

• Since the horizontal stretch changes both angles and lengths, neither property is preserved by the entire sequence.

</reasoning>

<answer>
<mcq-option>(A) Angle measures only</mcq-option>
<mcq-option>(B) Segment lengths only</mcq-option>
<mcq-option>(C) Both angle measures and segment lengths</mcq-option>
<mcq-correct>(D) Neither angle measures nor segment lengths</mcq-correct>
</answer>

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Answer:

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<reasoning>

Analyze the first two transformations

Using the Geometric Transformations and Rigid Transformations knowledge points

• A dilation about a point \(P\) preserves angle measures but changes segment lengths.
• A rotation about \(P\) is a rigid transformation, preserving both angle measures and segment lengths.

Analyze the horizontal stretch

• A horizontal stretch about a vertical line \(\overleftrightarrow{QR}\) scales coordinates in only one direction.
• This non-rigid transformation distorts shapes, changing both segment lengths and angle measures.

Determine what is preserved overall

• Since the horizontal stretch changes both angles and lengths, neither property is preserved by the entire sequence.

</reasoning>

<answer>
<mcq-option>(A) Angle measures only</mcq-option>
<mcq-option>(B) Segment lengths only</mcq-option>
<mcq-option>(C) Both angle measures and segment lengths</mcq-option>
<mcq-correct>(D) Neither angle measures nor segment lengths</mcq-correct>
</answer>

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