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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 ofdegrees clockwise about the origin, then translation of ()Remaining Attempts : 3

1 answer

To determine the sequence of transformations that demonstrates the congruence of the triangles in the figure, we can analyze the shifts that take place from one triangle to another.

Given the triangles:

  1. Triangle ABC with vertices A(2, -2), B(2, -4), and C(5, -4).
  2. Triangle in the second quadrant with vertices (-3, 6), (-5, 6), and (-5, 3).
  3. Triangle in the third quadrant with vertices (-2, -2), (-4, -2), and (-4, -5).

The arrows indicate transformations from the third triangle (in the third quadrant) to the second triangle (in the second quadrant).

Step 1: Analyze each arrow's transformation.

  • The arrow from (-2, -2) to (-3, 6) suggests a transformation that moves the point to a different location, indicating a combination of rotation and translation.

  • The arrow from (-4, -2) to (-5, 6) shows a similar movement of one triangle's vertex to the other triangle's vertex.

  • The arrow from (-4, -5) to (-5, 3) suggests the same type of transformation.

Step 2: Determine the type of transformations.

The movement of points generally indicates a rotation followed by a translation:

  1. Rotation of 180 degrees: When points in quadrant III are moved to quadrant II, it suggests a rotation around the origin. This would move points in quadrant III up into quadrant II.

  2. Translation: Following the rotation, additional translations are made to align the matching points of the triangles correctly.

Conclusion

Based on the observed transformations of the points:

  • Rotation of 180 degrees clockwise about the origin: This transformation would rotate the points in the third quadrant to their corresponding points in the second quadrant.
  • Translation of (-1, 1) (derived from examining how much the points shift after rotation to reach their targets).

So, the sequence of transformations that proves the congruence of the triangles is:

  • Rotation of 180 degrees clockwise about the origin, then translation of (-1, 1).