Question

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?

(1 point)
rotation of
degrees clockwise about the origin, then translation of (
)

Answer 1

To determine the sequence of transformations that proves the congruence of the triangles A, B, C (first triangle) and the one in the second quadrant, we need to analyze the positions of the triangles and the arrows between them.

The first triangle with vertices at \( A(2, -2) \), \( B(2, -4) \), and \( C(5, -4) \) is in the fourth quadrant. The second triangle with vertices at \( (-3, 6) \), \( (-5, 6) \), and \( (-5, 3) \) is in the second quadrant.

1. Rotation: It seems the first triangle needs to rotate to align it properly with the second triangle. By examining the coordinates, it looks like Triangle A, B, C must be rotated 180 degrees clockwise about the origin \( (0, 0) \). This will move the first triangle from the fourth quadrant to the second quadrant.

2. Translation: After the rotation, the triangle appears to shift to the vertices of the second triangle. The rotation of the triangle would position it somewhere around the point in the second quadrant, and a translation is necessary to map the rotated triangle's vertices to those of the second triangle. The final translation would move the triangle to appropriately match the vertices of the second triangle.

Considering the arrows and the coordinates given, the transformation can be summarized as follows:

• Rotation: 180 degrees clockwise about the origin.
• Translation: The translation vector would need to account from a point like \( (-5, 3) \) to \( (-3, 6) \) indicating a move of \( (+2, +3) \).

So the complete transformation description is:

Rotation of 180 degrees clockwise about the origin, then translation of (2, 3).