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Triangle PQR is translated by the rule [tex]\((x-1, y-1)\)[/tex] and then dilated by a scale factor of 3 centered at the origin. Which statement describes the properties of triangles PQR and [tex]\(P^\ \textless \ em\ \textgreater \ Q^\ \textless \ /em\ \textgreater \ R^\ \textless \ em\ \textgreater \ \)[/tex] after the transformations?

A. [tex]\(\angle P\)[/tex] and [tex]\(\angle P^\ \textless \ /em\ \textgreater \ \)[/tex] are congruent after the translation, but not after the dilation.
B. [tex]\(\angle P\)[/tex] and [tex]\(\angle P^*\)[/tex] are congruent after the dilation, but not after the translation.
C. [tex]\(\overline{PQ}\)[/tex] and [tex]\(\overline{P^{\prime\prime} Q^{\prime\prime}}\)[/tex] are congruent after the translation, but not after the dilation.
D. [tex]\(\overline{PQ}\)[/tex] and [tex]\(\overline{P^{\prime\prime} Q^{\prime\prime}}\)[/tex] are congruent after the dilation, but not after the translation.



Answer :

GinnyAnswer
Let's carefully analyze the transformations and the properties of the triangles:

### Translation by the Rule [tex]$(x-1, y-1)$[/tex]

1. Translation Properties:
- Angles: Translation is a rigid transformation, which means it preserves the distances and angles within the shape. So, after the translation, the angles of [tex]$\angle P$[/tex], [tex]$\angle Q$[/tex], and [tex]$\angle R$[/tex] are congruent to [tex]$\angle P^$[/tex], [tex]$\angle Q^$[/tex], and [tex]$\angle R^$[/tex] respectively.
- Lengths of Segments: Since the translation is rigid, the lengths of the segments are preserved. Therefore, the lengths [tex]$\overline{PQ}$[/tex] and [tex]$\overline{P'Q'}$[/tex] remain unchanged.

### Dilation with a Scale Factor of 3 Centered at the Origin

2. Dilation Properties:
- Angles: Dilation preserves angles. Therefore, [tex]$\angle P$[/tex], [tex]$\angle Q$[/tex], and [tex]$\angle R$[/tex] will be congruent to [tex]$\angle P^$[/tex], [tex]$\angle Q^$[/tex], and [tex]$\angle R^$[/tex] respectively even after the dilation.
- Lengths of Segments: Dilation changes the lengths of the segments by the factor of the scale factor. In this case, since the scale factor is 3, the lengths of segments will be multiplied by 3. Therefore, [tex]$\overline{PQ}$[/tex] will not be congruent to [tex]$\overline{P''Q''}$[/tex].

### Summary and Correct Statement
- Angles: Both translation and dilation preserve angles, making the statement about non-congruent angles incorrect.
- Lengths of Segments: Translation does not change the lengths of segments, but dilation does. Therefore, [tex]$\overline{PQ}$[/tex] and [tex]$\overline{P''Q''}$[/tex] are congruent after the translation but not after the dilation.

### Conclusion
The correct statement is:
> [tex]$\overline{PQ}$[/tex] and [tex]$\overline{P''Q''}$[/tex] are congruent after the translation, but not after the dilation.

Hence, the correct answer is:
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