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.

Question

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).