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Use the graph to answer the question.

graph of triangle ABC with vertices at negative 2 comma negative 7, 3 comma negative 2, 2 comma negative 10 and a second triangle A prime, B prime C prime with vertices at 2 comma 7, negative 3 comma 2, negative 2 comma 10

Determine the direction and degree of rotation used to create the image.
90° counterclockwise rotation
270° counterclockwise rotation
90° clockwise rotation
180° clockwise rotation

1 answer

To determine the direction and degree of rotation used to create the image of triangle ABC to triangle A'B'C', we need to analyze the coordinates of the vertices before and after the transformation.

The original triangle ABC has the vertices:

  • A(-2, -7)
  • B(3, -2)
  • C(2, -10)

The transformed triangle A'B'C' has the vertices:

  • A'(2, 7)
  • B'(-3, 2)
  • C'(-2, 10)

Steps to identify the transformation:

  1. Identify the Center of Rotation: For a rotation transformation, the center of rotation is often the origin (0,0).

  2. Calculating the rotation from A to A':

    • Original A = (-2, -7)
    • Transformed A' = (2, 7)

    To see the change in coordinates, we can observe:

    • The x-coordinate changes from -2 to 2 (which is +4).
    • The y-coordinate changes from -7 to 7 (which is +14).

  3. Rotating Points: A rotation of 90° counterclockwise can be described as transforming points (x, y) to (-y, x).

    • Applying this to A(-2, -7):
      • Rotate 90° counterclockwise: \[ (-(-7), -2) = (7, -2) \] This is not equal to A'(2, 7).

    A rotation of 180° results in:

    • Rotate 180°: \[ (-(-2), -(-7)) = (2, 7) \] This matches A'.

  4. Verifying other points: For B(3, -2):

    • Rotate 180°: \[ (-3, 2) \] This matches B'.

    For C(2, -10):

    • Rotate 180°: \[ (-2, 10) \] This matches C'.

Based on the transformations and checks:

Conclusion:

The rotation used to create the image from triangle ABC to triangle A'B'C' is a 180° rotation clockwise.