To determine which transformations could show that triangles \( \Delta ABC \) and \( \Delta DEF \) are similar, we start by calculating the lengths of the sides of each triangle.
For \( \Delta ABC \):
Vertices are \( A(0,3) \), \( B(0,1) \), and \( C(-4,2) \).
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\( AB \): \[ AB = \sqrt{(0 - 0)^2 + (3 - 1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2 \]
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\( BC \): \[ BC = \sqrt{(0 - (-4))^2 + (1 - 2)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
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\( CA \): \[ CA = \sqrt{(0 - (-4))^2 + (3 - 2)^2} = \sqrt{(4)^2 + (1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
So, the side lengths of \( \Delta ABC \) are \( AB = 2 \), \( BC = \sqrt{17} \), and \( CA = \sqrt{17} \).
For \( \Delta DEF \):
Vertices are \( D(0,6) \), \( E(0,2) \), and \( F(8,4) \).
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\( DE \): \[ DE = \sqrt{(0 - 0)^2 + (6 - 2)^2} = \sqrt{0 + 4^2} = 4 \]
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\( EF \): \[ EF = \sqrt{(0 - 8)^2 + (2 - 4)^2} = \sqrt{(-8)^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \]
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\( FD \): \[ FD = \sqrt{(0 - 8)^2 + (6 - 4)^2} = \sqrt{(-8)^2 + (2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \]
So, the side lengths of \( \Delta DEF \) are \( DE = 4 \), \( EF = 2\sqrt{17} \), and \( FD = 2\sqrt{17} \).
Comparing the Sides: The corresponding sides are:
- \( AB (= 2) \) corresponds to \( DE (= 4) \)
- \( BC (= \sqrt{17}) \) corresponds to \( EF (= 2\sqrt{17}) \)
- \( CA (= \sqrt{17}) \) corresponds to \( FD (= 2\sqrt{17}) \)
Now, let's check the ratios of the corresponding sides:
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Ratio of \( AB \) to \( DE \): \[ \frac{AB}{DE} = \frac{2}{4} = \frac{1}{2} \]
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Ratio of \( BC \) to \( EF \): \[ \frac{BC}{EF} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]
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Ratio of \( CA \) to \( FD \): \[ \frac{CA}{FD} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]
All ratios are equal to \( \frac{1}{2} \), indicating that \( \Delta ABC \) and \( \Delta DEF \) are similar by dilation with a scale factor of \( \frac{1}{2} \).
Now let’s analyze the transformations:
- Option A: Dilating by \( \frac{1}{4} \) does not yield a \( \frac{1}{2} \) similarity.
- Option B: Dilating by \( -2 \) would reverse the triangle and not maintain similarity.
- Option C: Dilating by \( 2 \) would enlarge the triangle too much, leading away from similarity.
- Option D: Dilating by \( \frac{1}{2} \) correctly matches the similarity criteria and maintains orientation.
Thus, the answer is:
Option D: \( \Delta ABC \) is dilated by a scale factor of \( \frac{1}{2} \) and is reflected across the x-axis.