Question

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 6 to 6 in increments of 1. Three triangles are formed by joining three plotted points each. The coordinates of the plotted points for the first triangle upper A upper B upper C joined by solid lines are upper A is left parenthesis negative 4 comma 5 right parenthesis, upper B is left parenthesis negative 1 comma 3 right parenthesis, and upper C is left parenthesis negative 3 comma 1 right parenthesis. The coordinates for the second triangle upper A prime upper B prime upper C prime joined by dotted lines are as follows: upper A prime at left parenthesis 4 comma 5 right parenthesis, upper B prime at left parenthesis 1 comma 3 right parenthesis, and upper C prime at left parenthesis 3 comma 1 right parenthesis. The coordinates of the plotted points for the third triangle upper A double prime upper B double prime upper C double prime joined by lines made of dashes and dots are as follows: upper A double prime at left parenthesis 1 comma 0 right parenthesis, upper B double prime at left parenthesis negative 2 comma negative 2 right parenthesis, and upper C double prime at left parenthesis 0 comma negative 4 right parenthesis.
How would you describe this series of transformations? Reflection across the y
-axis and then a translation of (−3,−5)
shows that triangle ABC
is congruent to triangle A′′B"C"
.
Reflection across the y -axis and then a translation of left parenthesis negative 3 comma negative 5 right parenthesis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Rotation of 90 degrees clockwise and then reflection across the x
-axis shows that triangle ABC
is congruent to triangle A′′B"C"
.
Rotation of 90 degrees clockwise and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Translation of (2,0)
and then reflection across the x
-axis shows that triangle ABC
is congruent to triangle A′′B"C"
.
Translation of left parenthesis 2 comma 0 right parenthesis and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Since triangles ABC
and A′′B"C"
do not have the same orientation, they are not congruent..

Answer 1

To analyze the transformations between the triangles \( ABC \), \( A'B'C' \), and \( A''B''C'' \), we consider their coordinates carefully.

### Triangle \( ABC \):
- \( A(-4, 5) \)
- \( B(-1, 3) \)
- \( C(-3, 1) \)

### Triangle \( A'B'C' \):
- \( A'(4, 5) \)
- \( B'(1, 3) \)
- \( C'(3, 1) \)

### Triangle \( A''B''C'' \):
- \( A''(1, 0) \)
- \( B''(-2, -2) \)
- \( C''(0, -4) \)

### Transformation Analysis:

#### 1. **Reflecting Triangle ABC Across the Y-Axis**:
The reflection of a point \( (x, y) \) across the y-axis results in \( (-x, y) \):
- \( A \) becomes \( (4, 5) \)
- \( B \) becomes \( (1, 3) \)
- \( C \) becomes \( (3, 1) \)

After reflection, we observe that the new coordinates match those of triangle \( A'B'C' \).

#### 2. **Checking Orientation**:
Since triangle \( ABC \) and triangle \( A'B'C' \) have the same orientation, they are congruent through the reflection across the y-axis.

#### 3. **Translation**:
Next, we can describe the transformation from \( A'B'C' \) to \( A''B''C'' \).
- The coordinates of the vertices in \( A'B'C' \) after translation of \( (-3, -5) \) will be:
- \( A'(4, 5) + (-3, -5) = (1, 0) \) (which is \( A'' \))
- \( B'(1, 3) + (-3, -5) = (-2, -2) \) (which is \( B'' \))
- \( C'(3, 1) + (-3, -5) = (0, -4) \) (which is \( C'' \))

Thus, the transformation is** a reflection across the y-axis** followed by **a translation of \((-3, -5)\)**, which confirms that triangle \( ABC \) is congruent to triangle \( A''B''C''\).

### Conclusion:
In conclusion, the statement that best encapsulates these transformations is:
**Reflection across the y-axis and then a translation of \((-3, -5)\) shows that triangle ABC is congruent to triangle \( A''B''C'' \)**.

Therefore the best answer among the options provided is:
- **Reflection across the y-axis and then a translation of (-3, -5) shows that triangle ABC is congruent to triangle A''B''C''**.