Question

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Two triangles, upper A upper B upper C and upper E upper G upper H, are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis negative 6 comma negative 2 right parenthesis, upper B left parenthesis negative 4 comma negative 5 right parenthesis, and upper C left parenthesis negative 1 comma negative 5 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis 4 comma 6 right parenthesis, upper G left parenthesis 7 comma 6 right parenthesis, and upper H left parenthesis 2 comma 3 right parenthesis. In triangle upper E upper G upper H, side upper E upper G is labeled as h, side upper G upper H is labeled as e, and side upper H upper E is labeled as g.

what is the sequence of transformations needed to move triangle abc to triangle EGH to show that the triangles are congruent?

reflection over the x-axis, then a translation of (__)

Answer 1

To determine the sequence of transformations needed to move triangle ABC to triangle EGH and show that they are congruent, we can analyze the coordinates of the vertices of both triangles.

### Triangle ABC:
- A: (-6, -2)
- B: (-4, -5)
- C: (-1, -5)

### Triangle EGH:
- E: (4, 6)
- G: (7, 6)
- H: (2, 3)

### Step 1: Reflection
Looking at the coordinates, we notice that triangle ABC is located in the third quadrant (where both x and y are negative), while triangle EGH is in the first quadrant (where both x and y are positive).

To move triangle ABC into the first quadrant, we first reflect it over the x-axis. This reflection will change the signs of the y-coordinates of each point in triangle ABC:
- A: (-6, -2) becomes A': (-6, 2)
- B: (-4, -5) becomes B': (-4, 5)
- C: (-1, -5) becomes C': (-1, 5)

### Step 2: Translation
Next, to align triangle A'B'C' with triangle EGH, we need to translate the vertices.

The vertices after reflection A', B', and C' are:
- A': (-6, 2)
- B': (-4, 5)
- C': (-1, 5)

The translation needed takes A' to E (4, 6). We calculate the translation vector by finding the shift required:
- Movement in the x-direction: 4 - (-6) = 4 + 6 = 10
- Movement in the y-direction: 6 - 2 = 4

Thus, we can express the translation in vector form as (10, 4).

### Conclusion
The sequence of transformations needed to move triangle ABC to triangle EGH is:
- Reflection over the x-axis,
- Then, a translation of (10, 4).

So, the answer is:
**reflection over the x-axis, then a translation of (10, 4)**.