To demonstrate that triangle ABC is congruent to triangle GHE through a series of transformations (without a reflection over a diagonal axis), we can follow these steps:
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Translation: First, we will translate triangle ABC so that point A moves to the location of point G.
- The coordinates of A are (1, -3) and we want to move it to G at (-2, 2).
- The translation vector can be found by subtracting the coordinates of A from G: \[ \text{Translation vector} = (-2 - 1, 2 - (-3)) = (-3, 5). \]
- Therefore, we translate triangle ABC by (-3, 5). After this translation, the new coordinates of the triangle will be:
- A' = (1 - 3, -3 + 5) = (-2, 2)
- B' = (1 - 3, -2 + 5) = (-2, 3)
- C' = (4 - 3, -3 + 5) = (1, 2)
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Rotation: Next, we need to rotate the translated triangle A'B'C' to align with triangle GHE. We notice before transformation:
- G is at (-2, 2), H is at (-4, 2), and E is at (-2, 5).
- We will perform a 90-degree clockwise rotation about point G (-2, 2).
- To apply this rotation to each point:
- For a point (x, y), a 90-degree clockwise rotation around point (h, k) is given by: \[ (x', y') = (k - (y - k), h + (x - h)). \]
- For B' (-2, 3): \[ (3 - 2, -2 - (-2)) = (-2, -2). \]
- For C' (1, 2): \[ (2 - (2 - 2), -2 + (1 - (-2))) = (2, -1). \]
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Verifying Rotation: After rotation:
- Point A' becomes G = (-2, 2).
- Point B' becomes H = (-4, 2).
- Point C' becomes E = (-2, 5).
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Conclusion: After executing the translation followed by a rotation, we arrive at triangle GHE exactly aligning with the transformed triangle ABC. Thus, triangle ABC can be congruently transformed to triangle GHE through a translation and a rotation, proving the congruence without any reflections.
The final position of triangles confirms their congruence:
- Triangle ABC is located at A'(-2, 2), B'(-4, 2), C'(-2, 5).
- Triangle GHE is located at G(-2, 2), H(-4, 2), E(-2, 5).
Thus, the congruence can be achieved through translation followed by rotation without any reflection over a diagonal axis.