To show that triangle ABC is congruent to triangle GHE through a sequence of transformations, we can follow these steps:
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Translation: First, we will translate triangle ABC to a new position to align one point of triangle ABC with a corresponding point of triangle GHE. We can translate triangle ABC by moving it to the left and up to match point A with point G.
- Point A (1, -3) translates to G (-2, 2).
- To move from A to G, we find the translation vector:
- \( \Delta x = -2 - 1 = -3 \)
- \( \Delta y = 2 - (-3) = 5 \)
- So the translation vector is (-3, 5).
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Apply the translation: Now, we translate all points of triangle ABC by the vector (-3, 5):
- A' (1, -3) becomes A'' (-2, 2).
- B' (1, -2) becomes B'' (-2, 3).
- C' (4, -3) becomes C'' (1, 2).
After the translation, triangle A''B''C'' has the vertices:
- A'' (-2, 2)
- B'' (-2, 3)
- C'' (1, 2)
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Reflection: Next, we need to reflect triangle A''B''C'' to match the orientation of triangle GHE. Triangle GHE has points:
- G (-2, 2)
- H (-4, 2)
- E (-2, 5)
Reflecting triangle A''B''C'' across the line x = -2, the transformation will adjust the coordinates of points B'' and C'' as follows:
- A'' will remain unchanged at (-2, 2).
- B'' (-2, 3) reflects to (-4, 3).
- C'' (1, 2) reflects to (-5, 2) (Note: Change in x-value since it's a reflection across a vertical line).
However, note that we need to ensure the corresponding points match, specifically (H should correspond to B'' and E to C'').
Through a combination of translation and reflection, we can show that:
- Translate -3 units left and 5 units up.
- Reflect left across x = -2.
Conclusion:
Thus, triangle ABC can be transformed into triangle GHE by first translating it and then reflecting it across the line x = -2. Hence, triangles ABC and GHE are congruent through these transformations.