Question

which composition of similarity transformations maps \\(\delta lmn\\) to \\(\delta lmn\\)?

a dilation with a scale factor less than 1 and then a reflection
a dilation with a scale factor less than 1 and then a translation
a dilation with a scale factor greater than 1 and then a reflection
a dilation with a scale factor greater than 1 and then a translation

Explanation:

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<reasoning>

Determine the scale factor of the dilation

\[

$$\begin{aligned} &LN = 2,\quad L'N' = 4\\ &k = \frac{L'N'}{LN} = \frac{4}{2} = 2 > 1 \end{aligned}$$

\]

Identify the second transformation

\[

$$\begin{aligned} &\text{Orientation of }\Delta LMN\text{ is counterclockwise: } L(-1,3) \to M(3,1) \to N(-1,1)\\ &\text{Orientation of }\Delta L'M'N'\text{ is clockwise: } L'(-2,-1) \to M'(6,-5) \to N'(-2,-5)\\ &\text{A change in orientation indicates a reflection.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a dilation with a scale factor less than 1 and then a reflection</mcq-option>
<mcq-option>a dilation with a scale factor less than 1 and then a translation</mcq-option>
<mcq-correct>a dilation with a scale factor greater than 1 and then a reflection</mcq-correct>
<mcq-option>a dilation with a scale factor greater than 1 and then a translation</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
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]
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</post_analysis>

Answer:

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<reasoning>

Determine the scale factor of the dilation

\[

$$\begin{aligned} &LN = 2,\quad L'N' = 4\\ &k = \frac{L'N'}{LN} = \frac{4}{2} = 2 > 1 \end{aligned}$$

\]

Identify the second transformation

\[

$$\begin{aligned} &\text{Orientation of }\Delta LMN\text{ is counterclockwise: } L(-1,3) \to M(3,1) \to N(-1,1)\\ &\text{Orientation of }\Delta L'M'N'\text{ is clockwise: } L'(-2,-1) \to M'(6,-5) \to N'(-2,-5)\\ &\text{A change in orientation indicates a reflection.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a dilation with a scale factor less than 1 and then a reflection</mcq-option>
<mcq-option>a dilation with a scale factor less than 1 and then a translation</mcq-option>
<mcq-correct>a dilation with a scale factor greater than 1 and then a reflection</mcq-correct>
<mcq-option>a dilation with a scale factor greater than 1 and then a translation</mcq-option>
</answer>

<post_analysis>
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"Mathematics",
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"Similarity Transformations"
]
}
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