To determine which transformations could be used to show that triangles ΔABC and ΔDEF are similar, we need to first analyze their coordinates and the potential transformations.
The vertices of ΔABC are:
- A(0, 3)
- B(0, 1)
- C(−4, 2)
The vertices of ΔDEF are:
- D(0, 6)
- E(0, 2)
- F(8, 4)
Step 1: Compare the side lengths of the triangles.
For ΔABC:
- Length AB = |3 - 1| = 2
- Length BC = √((-4 - 0)² + (2 - 1)²) = √(16 + 1) = √17
- Length AC = √((-4 - 0)² + (2 - 3)²) = √(16 + 1) = √17
So the side lengths of ΔABC: 2, √17, √17.
For ΔDEF:
- Length DE = |6 - 2| = 4
- Length EF = √((8 - 0)² + (4 - 2)²) = √(64 + 4) = √68
- Length DF = √((8 - 0)² + (4 - 6)²) = √(64 + 4) = √68
So the side lengths of ΔDEF: 4, √68, √68.
Step 2: Determine any scale factors.
We can observe that the ratios of corresponding sides could indicate similarity. For similarity, we can take the ratio of corresponding side lengths:
\[ \text{From } AB \text{ to } DE: \frac{2}{4} = \frac{1}{2} \]
\[ \text{From } AC \text{ (or } BC\text{) to } DF \text{ (or } EF\text{): } \frac{\sqrt{17}}{\sqrt{68}} = \frac{\sqrt{17}}{2\sqrt{17}} = \frac{1}{2} \]
Both pairs of corresponding sides have the same ratio of 1:2.
Step 3: Identify potential transformations.
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Dilation of ΔABC by a scale factor of \( \frac{1}{2} \) would reduce the size of the triangle proportionally, which corresponds to the scaling we observed from triangle ABC to DEF.
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Reflection across the x-axis or y-axis is also included in the possible transformations.
After analyzing the possible transformations from the given options, the correct response that fits the determined transformations would be:
ΔABC is dilated by a scale factor of \( \frac{1}{2} \) and is reflected across the x-axis.
Therefore, we conclude that this transformation would show that the triangles are similar.
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