Question

Triangle ABC is a triangle on a graph. A is at (1, -3), B is at (1, -2), and C is at (4, -3). Another triangle is on the graph, triangle GHE. G is at (-2, 2), H is at (-4, 2), and E is at (-2, 5).

ABC is congruent to GHE. Describe a sequence of transformations from ABC to GHE that proves their congruence without a reflection over a diagonal axis

Answer 1

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:

1. **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)

2. **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):
\[
(2 - (2 - 2), -2 + (1 - (-2))) = (2, -1).
\]

3. **Verifying Rotation**: After rotation:
- Point A' becomes G = (-2, 2).
- Point B' becomes H = (-4, 2).
- Point C' becomes E = (-2, 5).

4. **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.