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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?

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

1 answer

To determine the sequence of transformations that proves the congruence of the triangles, let's analyze the triangles and the arrows indicating the movement of the points.

  1. Identifying the coordinates:

    • Triangle ABC has vertices:

      • A(2, -2)
      • B(2, -4)
      • C(5, -4)

    • The triangle in the second quadrant has vertices:

      • D(-3, 6)
      • E(-5, 6)
      • F(-5, 3)

    • The triangle in the third quadrant has vertices:

      • G(-2, -2)
      • H(-4, -2)
      • I(-4, -5)

  2. Analyzing transformations:

    • First, look at the arrow from G(-2, -2) to D(-3, 6). This indicates a movement that suggests a rotation might have occurred.
    • The next arrow from H(-4, -2) to E(-5, 6) indicates a similar movement, and it follows the same pattern.
    • The final arrow from I(-4, -5) to F(-5, 3) shows that this point was also moved according to the previous transformation.

  3. Determining the rotation:

    • The points in Triangle ABC are all transformed to a new position in the second triangle through rotation.
    • Analyzing the changes in coordinates can help determine the angle of the rotation.

From visualizing the points and their movements, it appears that all points might have been rotated around the origin.

Supposing the triangles are congruent, and the overall movement leads them to positions satisfying congruency:

Sequence of transformations:

  • Rotation: 90 degrees clockwise about the origin will transform Triangle ABC such that point A(2, -2) goes to D(-3, 6), point B(2, -4) goes to point E(-5, 6), and point C(5, -4) goes to point F(-5, 3).

  • Translation: After the rotation, the triangle may require a translation to properly align with the vertices of Triangle DEF. This can be determined by the shift in coordinates.

Looking closely at the translations, if we move from the new rotated positions to the second triangle’s coordinates, this translates each point downward or to the left by a certain amount.

In this case, we can express it as:

  • Translation of (-1, 0) after the rotation.

Putting this all together, we can conclude:

The transformations are:

  • Rotation of 90 degrees clockwise about the origin, then translation of (-1, 0).