To determine the transformation that shows congruence between triangle ABC (with vertices A(2, 1), B(7, 1), C(4, 3)) and triangle A'B'C' (with vertices A'(-1, 2), B'(-1, 7), C'(-3, 4)), we can analyze the coordinates of both triangles.
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Locate Triangle ABC:
- A(2, 1)
- B(7, 1)
- C(4, 3)
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Locate Triangle A'B'C':
- A'(-1, 2)
- B'(-1, 7)
- C'(-3, 4)
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Determine the Relationship:
- Triangle ABC is positioned in the first quadrant while triangle A'B'C' is positioned in the second quadrant.
- We can use a combination of transformations (rotations and reflections) and check if either achieve congruence.
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Testing Rotations:
- A 90-degree counterclockwise rotation around the origin transforms point (x, y) to (-y, x).
- Applying a 90-degree counterclockwise rotation to the vertices of triangle ABC:
- A(2, 1) transforms to (-1, 2),
- B(7, 1) transforms to (-1, 7),
- C(4, 3) transforms to (-3, 4).
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Results: After this transformation, we achieve the vertices of triangle A'B'C'.
So, the transformation that shows congruence between triangle ABC and triangle A'B'C' is indeed a 90-degree counterclockwise rotation around the origin.