Use the image to answer the question.

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).