Let's analyze the transformations step by step to determine how they affect the congruence of the triangles.
### Step 1: Translation
The translation rule is [tex]\((x-1, y-1)\)[/tex]. This rule shifts every point of triangle [tex]\( PQR \)[/tex] by 1 unit to the left and 1 unit down. Let's denote the original points as [tex]\( P(Px, Py) \)[/tex], [tex]\( Q(Qx, Qy) \)[/tex], and [tex]\( R(Rx, Ry) \)[/tex]. After the translation:
- [tex]\( P \)[/tex] moves to [tex]\( P_t = (Px-1, Py-1) \)[/tex].
- [tex]\( Q \)[/tex] moves to [tex]\( Q_t = (Qx-1, Qy-1) \)[/tex].
- [tex]\( R \)[/tex] moves to [tex]\( R_t = (Rx-1, Ry-1) \)[/tex].
The translation does not change the shape or size of the triangle - it only changes its position. Hence, triangle [tex]\( PQR \)[/tex] is congruent to triangle [tex]\( P_tQ_tR_t \)[/tex].
### Step 2: Dilation
The dilation rule is to scale by a factor of 3 centered at the origin. Dilation by a scale factor of [tex]\( k=3 \)[/tex] means each point's coordinates are multiplied by 3:
- [tex]\( P_t = (Px-1, Py-1) \)[/tex] dilates to [tex]\( (3(Px-1), 3(Py-1)) = (3Px-3, 3Py-3) \)[/tex].
- [tex]\( Q_t = (Qx-1, Qy-1) \)[/tex] dilates to [tex]\( (3(Qx-1), 3(Qy-1)) = (3Qx-3, 3Qy-3) \)[/tex].
- [tex]\( R_t = (Rx-1, Ry-1) \)[/tex] dilates to [tex]\( (3(Rx-1), 3(Ry-1)) = (3Rx-3, 3Ry-3) \)[/tex].
While dilation does preserve the shape of the triangle, it does not preserve its size. The new triangle [tex]\( P'(3Px-3, 3Py-3) \)[/tex], [tex]\( Q'(3Qx-3, 3Qy-3) \)[/tex], and [tex]\( R'(3Rx-3, 3Ry-3) \)[/tex] is similar to [tex]\( PQR \)[/tex], but it is not congruent to [tex]\( PQR \)[/tex] because its sides are three times longer.
### Conclusion
From the above analysis, we can draw the following conclusions:
1. Triangle [tex]\( PQR \)[/tex] is congruent to triangle [tex]\( P_tQ_tR_t \)[/tex] after the translation, but not after the dilation because the sides are scaled by a factor of 3.
2. The triangles are similar but not congruent after the dilation because the dilation alters the size of the triangle.
Given these points, the correct statement is:
- [tex]\( PQR \)[/tex] and [tex]\( P'Q'R' \)[/tex] are congruent after the translation but not after the dilation.
Thus, the correct statement is:
[tex]\[ \text{P and P' are congruent after the translation but not after the dilation.} \][/tex]
