To determine which statement is true about the dilation of triangle [tex]\( ABC \)[/tex] to triangle [tex]\( A'B'C' \)[/tex], we need to find the correct scale factor that transforms the coordinates [tex]\( A(2,4) \)[/tex], [tex]\( B(4,0) \)[/tex], and [tex]\( C(6,6) \)[/tex].
We can find the dilated coordinates [tex]\( A'(x', y') \)[/tex], [tex]\( B'(x', y') \)[/tex], and [tex]\( C'(x', y') \)[/tex] using different scale factors and compare them to the original vertices.
Let's try each given scale factor and see which one accurately scales the triangle [tex]\( ABC \)[/tex].
1. Scale factor of 0.5:
- [tex]\( A'(2 \times 0.5, 4 \times 0.5) = (1, 2) \)[/tex]
- [tex]\( B'(4 \times 0.5, 0 \times 0.5) = (2, 0) \)[/tex]
- [tex]\( C'(6 \times 0.5, 6 \times 0.5) = (3, 3) \)[/tex]
2. Scale factor of 1.5:
- [tex]\( A'(2 \times 1.5, 4 \times 1.5) = (3, 6) \)[/tex]
- [tex]\( B'(4 \times 1.5, 0 \times 1.5) = (6, 0) \)[/tex]
- [tex]\( C'(6 \times 1.5, 6 \times 1.5) = (9, 9) \)[/tex]
3. Scale factor of 2:
- [tex]\( A'(2 \times 2, 4 \times 2) = (4, 8) \)[/tex]
- [tex]\( B'(4 \times 2, 0 \times 2) = (8, 0) \)[/tex]
- [tex]\( C'(6 \times 2, 6 \times 2) = (12, 12) \)[/tex]
4. Scale factor of 2.5:
- [tex]\( A'(2 \times 2.5, 4 \times 2.5) = (5, 10) \)[/tex]
- [tex]\( B'(4 \times 2.5, 0 \times 2.5) = (10, 0) \)[/tex]
- [tex]\( C'(6 \times 2.5, 6 \times 2.5) = (15, 15) \)[/tex]
Next, we compare these scaled coordinates with the vertices of [tex]\( A'B'C' \)[/tex].
The only set of calculations that leads to a practical dilation that fits the problem's context is the scale factor of 2:
- [tex]\( A'(4, 8) \)[/tex]
- [tex]\( B'(8, 0) \)[/tex]
- [tex]\( C'(12, 12) \)[/tex]
Therefore, the correct statement is:
[tex]\[ \text{Triangle } ABC \text{ is dilated by a scale factor of } 2. \][/tex]
