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. not 10, 4

Answer 1

To transform triangle \(ABC\) to triangle \(EGH\) and demonstrate that they are congruent, let's first analyze their vertices:

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

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

### Step 1: Reflection over the x-axis
Reflect triangle \(ABC\) over the x-axis. This changes the sign of the y-coordinates of each vertex:

- Reflected \(A' ( -6, 2)\)
- Reflected \(B' ( -4, 5)\)
- Reflected \(C' ( -1, 5)\)

### Step 2: Translation
Now we need to translate the reflected triangle \(A'B'C'\) to get it to match triangle \(EGH\). We will calculate the translation needed.

The second half of the transformation involves translating each vertex of triangle \(A'B'C'\) so that it aligns with the corresponding vertices in triangle \(EGH\):

1. To translate \(A' (-6, 2)\) to \(E (4, 6)\):
\[
\text{Change in x} = 4 - (-6) = 10
\]
\[
\text{Change in y} = 6 - 2 = 4
\]
Thus, we need to translate by \((10, 4)\).

2. To verify:
- Translating \(B' (-4, 5)\) by \((10, 4)\):
\[
B'' = (-4 + 10, 5 + 4) = (6, 9)
\]
- Translating \(C' (-1, 5)\) by \((10, 4)\):
\[
C'' = (-1 + 10, 5 + 4) = (9, 9)
\]

### Final Transformation Sequence
1. Reflect triangle \(ABC\) over the x-axis.
2. Translate the reflected triangle by \((10, 4)\).

In summary,
- Reflect triangle \(ABC\) over the x-axis, then translate by \((10, 4)\) to obtain triangle \(EGH\) which demonstrates that the triangles are congruent.