To determine the congruence transformation used on triangle \( ABC \) that resulted in triangle \( DEF \), we can first find the transformations needed by computing the vectors between the points of both triangles.
Given:
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Points of triangle \( ABC \):
- A: (-1, 3)
- B: (-2, -1)
- C: (1, 0)
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Points of triangle \( DEF \):
- D: (3, -1)
- E: (4, 3)
- F: (1, 2)
Next, let's analyze the transformations step by step.
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Shifting the points:
Let's determine the translation that would move \( A \) to \( D \): \[ D - A = (3 - (-1), -1 - 3) = (4, -4) \]The shift vector from \( A \) to \( D \) is \( (4, -4) \). We need to see if the other points can be made to match after this shift:
- For \( B \): \[ D - B = (3 - (-2), -1 - (-1)) = (5, 0) \]
- For \( C \): \[ D - C = (3 - 1, -1 - 0) = (2, -1) \]
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Determining if there is a rotation: So a simple shift does not seem to make the three points match. We should check if the next transformation involves rotation.
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Testing with rotations: Let's test a \( 180^\circ \) rotation around the origin and then see if any translations match.
For a \( 180^\circ \) rotation about the origin, the transformation can be represented as: \[ (x, y) \to (-x, -y) \]
- Rotate \( A (-1, 3) \): \[ A' = (1, -3) \]
- Rotate \( B (-2, -1) \): \[ B' = (2, 1) \]
- Rotate \( C (1, 0) \): \[ C' = (-1, 0) \]
Now, shift all rotated points to try to match those in \( DEF \):
- Translating \( A' \) to match \( D(3, -1) \): \[ D - (1, -3) = (2, 2) \]
- For \( E \) we will check: \[ E - (2, 1) = (2, 2) \]
- For \( F \): \[ F - (-1, 0) = (2, 2) \]
These match with a translation \( (2, 2) \).
Thus, the transformations are:
- Rotate \( \triangle ABC \) \( 180^\circ \) counterclockwise.
- Shift it 4 units right and 2 units up.
The final answer corresponds most closely with:
Triangle \( ABC \) was reflected across the y-axis, then shifted 2 units to the right and 2 units up.