To find the coordinates of [tex]\( U^{\prime} \)[/tex] after the transformation [tex]\( \left(T_{<-3,1>} \circ D_4\right)(\triangle T U V) \)[/tex], we need to first apply the dilation [tex]\( D_4 \)[/tex] and then the translation [tex]\( T_{<-3,1>} \)[/tex]. Let's break it down step by step:
### 1. Dilation by a factor of 4
For the dilation, we multiply each coordinate of [tex]\( U \)[/tex] by the dilation factor of 4.
Given the original coordinates of [tex]\( U \)[/tex] as [tex]\( U(-8, 3) \)[/tex]:
- The x-coordinate after dilation: [tex]\( -8 \times 4 = -32 \)[/tex]
- The y-coordinate after dilation: [tex]\( 3 \times 4 = 12 \)[/tex]
So, after applying the dilation [tex]\( D_4 \)[/tex], the new coordinates of [tex]\( U \)[/tex] are:
[tex]\[ U_{\text{dilated}} = (-32, 12) \][/tex]
### 2. Translation by the vector [tex]\( \langle -3, 1 \rangle \)[/tex]
Next, we add the translation vector [tex]\( \langle -3, 1 \rangle \)[/tex] to the dilated coordinates:
- The x-coordinate after translation: [tex]\( -32 + (-3) = -35 \)[/tex]
- The y-coordinate after translation: [tex]\( 12 + 1 = 13 \)[/tex]
So, after applying the translation [tex]\( T_{<-3,1>} \)[/tex], the transformed coordinates of [tex]\( U \)[/tex] are:
[tex]\[ U^{\prime} = (-35, 13) \][/tex]
### Summary
After the transformation [tex]\( \left(T_{<-3,1>} \circ D_4\right) \)[/tex]:
- The coordinates of the dilated point [tex]\( U \)[/tex] are [tex]\( (-32, 12) \)[/tex].
- The final coordinates of [tex]\( U^{\prime} \)[/tex] after the full transformation are [tex]\( (-35, 13) \)[/tex].
