To determine if triangles \( \Delta ABC \) and \( \Delta DEF \) are similar, we can analyze their sides and check if a transformation can be applied to \( \Delta ABC \) to match \( \Delta DEF \).
Vertices:
-
Triangle \( \Delta ABC \):
- \( A(0, 3) \)
- \( B(0, 1) \)
- \( C(-4, 2) \)
-
Triangle \( \Delta DEF \):
- \( D(0, 6) \)
- \( E(0, 2) \)
- \( F(8, 4) \)
Step 1: Calculating Side Lengths
-
For \( \Delta ABC \):
- Side \( AB \): \( \sqrt{(0 - 0)^2 + (3 - 1)^2} = \sqrt{0 + 4} = 2 \)
- Side \( BC \): \( \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \)
- Side \( CA \): \( \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \)
-
For \( \Delta DEF \):
- Side \( DE \): \( \sqrt{(0 - 0)^2 + (6 - 2)^2} = \sqrt{0 + 16} = 4 \)
- Side \( EF \): \( \sqrt{(0 - 8)^2 + (2 - 4)^2} = \sqrt{(8)^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} \)
- Side \( FD \): \( \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{(8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} \)
Step 2: Comparing Ratios of Sides
Now we compare the ratios of corresponding sides to see if they are proportional:
- Compare \( \frac{AB}{DE} = \frac{2}{4} = \frac{1}{2} \)
- Compare \( \frac{BC}{EF} = \frac{\sqrt{17}}{\sqrt{68}} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \)
- Compare \( \frac{CA}{FD} = \frac{\sqrt{17}}{\sqrt{68}} = \frac{1}{2} \)
Step 3: Transformations
Since all the side ratios are the same (\( \frac{1}{2} \)) and triangles share the same angles due to proportional sides, the triangles are similar.
If \( \Delta ABC \) is dilated by a factor of 2 to become \( \Delta DEF \):
- The scaling is also consistent. Thus:
- \( \Delta ABC \) can be dilated by a factor of 2 and does not require reflection to align with \( \Delta DEF \).
Conclusion
The correct transformation is: ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis.
Thus, the answer is:
- ΔABC is dilated by a scale factor of 2 and is reflected across the y-axis.