Question

Use the image to answer the 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. Three triangles 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 2 comma negative 2 right parenthesis, upper B left parenthesis 2 comma negative 4 right parenthesis, and upper C left parenthesis 5 comma negative 4 right parenthesis. The second triangle, placed in the second quadrant, has its vertices marked with closed points at left parenthesis negative 3 comma 6 right parenthesis, left parenthesis negative 5 comma 6 right parenthesis, and left parenthesis negative 5 comma 3 right parenthesis. The third triangle, placed in the third quadrant, has its vertices marked with closed points at left parenthesis negative 2 comma negative 2 right parenthesis, left parenthesis negative 4 comma negative 2 right parenthesis, and left parenthesis negative 4 comma negative 5 right parenthesis. An arrow goes from the point left parenthesis negative 2 comma negative 2 right parenthesis to left parenthesis negative 3 comma 6 right parenthesis; an arrow goes from the point left parenthesis negative 4 comma negative 2 right parenthesis to left parenthesis negative 5 comma 6 right parenthesis; an arrow goes from left parenthesis negative 4 comma negative 5 right parenthesis to left parenthesis negative 5 comma 3 right parenthesis.

What sequence of transformations is depicted in the figure, proving the congruence of the triangles?

(2 points)
rotation of (___) degrees clockwise about the origin, then translation of (__, __​​​​​​​)​​​​​​​

Answer 1

To analyze the transformation that shows the congruence of the triangles, we look at the position of the triangles in the coordinate plane.

1. **Rotation**: The arrows indicate that there is a change in orientation from the triangle in the third quadrant to the triangle in the second quadrant. The transformation appears to rotate the triangle. Looking specifically at the points:
- Triangle ABC has points A(2, -2), B(2, -4), and C(5, -4).
- Triangle in second quadrant has points (-3, 6), (-5, 6), and (-5, 3).

From the coordinates, we can derive the necessary rotation. The triangle ABC is transformed from the third quadrant to the second quadrant, suggesting a clockwise rotation.

By analyzing the coordinates, it seems we need to rotate the triangle approximately **90 degrees clockwise**, which will move points from the third quadrant to the second quadrant.

2. **Translation**: After rotation, the triangles are aligned but may not occupy the same space. We must identify the translation that aligns these two transformed triangles. Looking at the movement from point (-2, -2) to (-3, 6) indicates a shift. If we analyze the horizontal and vertical shifts:
- The movement from x = -2 to x = -3 is a shift left by 1 unit.
- The movement from y = -2 to y = 6 is a shift up by 8 units.

Putting this together, the triangle moves:
- Left 1 unit (x coordinate decreased by 1)
- Up 8 units (y coordinate increased by 8)

Thus, the translation vector is (-1, 8).

### Final Answer
The sequence of transformations is:
- **rotation of 90 degrees clockwise about the origin, then translation of (-1, 8)**.