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?

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

Answer 1

To move triangle ABC to triangle EGH and show that the triangles are congruent, we can follow these steps:

1. **Reflection over the x-axis**: Reflect triangle ABC across the x-axis. This will change the y-coordinates of the vertices of triangle ABC from negative to positive:

- A(-6, -2) reflects to A'(-6, 2)
- B(-4, -5) reflects to B'(-4, 5)
- C(-1, -5) reflects to C'(-1, 5)

After reflection, the new vertices of triangle A'B'C' will be A'(-6, 2), B'(-4, 5), and C'(-1, 5).

2. **Translation**: Now, we need to translate triangle A'B'C' to the position of triangle EGH. Triangle EGH has vertices E(4, 6), G(7, 6), and H(2, 3). We can find the translation by determining how much we need to move each point.

For example, we can take the point A'(-6, 2) and see how to translate it to the point E(4, 6).
The translation vector can be found by calculating the difference in coordinates:

\[
\text{Translation vector} = (4 - (-6), 6 - 2) = (10, 4)
\]

So, you would translate all the points of A'B'C' by this vector (10, 4).

Thus, the sequence of transformations is:
**Reflection over the x-axis**, then **Translation of (10, 4)**.