Answer :
To determine the scale factor of the dilation, let's follow these steps methodically.
1. Determine the initial length of side [tex]\( \overline{AB} \)[/tex] of rectangle [tex]\( ABCD \)[/tex]:
The coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A(1, 1) \)[/tex] and [tex]\( B(4, 1) \)[/tex], respectively. The length of [tex]\( \overline{AB} \)[/tex] is the horizontal distance between these two points.
[tex]\[ \text{Length of } \overline{AB} = B_x - A_x = 4 - 1 = 3 \][/tex]
2. Find the length of the transformed side [tex]\( \overline{A'B'} \)[/tex] of rectangle [tex]\( A'B'C'D' \)[/tex]:
The problem states that the length of [tex]\( \overline{A'B'} = 6 \)[/tex].
3. Calculate the scale factor of the dilation:
The scale factor of a dilation is the ratio of the length of the transformed side to the length of the original side. Therefore,
[tex]\[ \text{Scale factor} = \frac{\text{Length of } \overline{A'B'}}{\text{Length of } \overline{AB}} = \frac{6}{3} = 2 \][/tex]
So, the scale factor of the dilation is [tex]\( 2 \)[/tex].
Thus, the correct answer is:
C. 2
1. Determine the initial length of side [tex]\( \overline{AB} \)[/tex] of rectangle [tex]\( ABCD \)[/tex]:
The coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A(1, 1) \)[/tex] and [tex]\( B(4, 1) \)[/tex], respectively. The length of [tex]\( \overline{AB} \)[/tex] is the horizontal distance between these two points.
[tex]\[ \text{Length of } \overline{AB} = B_x - A_x = 4 - 1 = 3 \][/tex]
2. Find the length of the transformed side [tex]\( \overline{A'B'} \)[/tex] of rectangle [tex]\( A'B'C'D' \)[/tex]:
The problem states that the length of [tex]\( \overline{A'B'} = 6 \)[/tex].
3. Calculate the scale factor of the dilation:
The scale factor of a dilation is the ratio of the length of the transformed side to the length of the original side. Therefore,
[tex]\[ \text{Scale factor} = \frac{\text{Length of } \overline{A'B'}}{\text{Length of } \overline{AB}} = \frac{6}{3} = 2 \][/tex]
So, the scale factor of the dilation is [tex]\( 2 \)[/tex].
Thus, the correct answer is:
C. 2