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A path between 3 cities is drawn on a map, creating the outlined region shown in the graph below.

The path between 3 cities forms a triangle. City A is at (2, 2), City B is at (4, 8), and City C is at (8, 4).

Which sequence of transformations would result in mapping the region onto itself?

A.
a rotation 90-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin

B.
a rotation 180-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin

C.
a rotation 90-degrees counterclockwise about the origin and then a rotation 180-degrees counterclockwise about the origin

D.
a reflection across the y-axis and then a reflection across the x-axis

1 answer

To determine which sequence of transformations maps the triangle formed by the cities A(2, 2), B(4, 8), and C(8, 4) onto itself, we can analyze the transformations provided in the options.

  1. Rotation 90 degrees counterclockwise about the origin:

    • New coordinates:
      • A' = (-2, 2)
      • B' = (-8, 4)
      • C' = (-4, 8)

    The shape of the triangle has changed; so it does not map onto itself.

  2. Rotation 270 degrees counterclockwise about the origin:

    • A (2, 2) --> A' (-2, -2)
    • B (4, 8) --> B' (-8, -4)
    • C (8, 4) --> C' (-4, -8)

    The shape of the triangle has changed, so it does not map onto itself.

  3. Rotation 180 degrees counterclockwise about the origin:

    • A (2, 2) --> A' (-2, -2)
    • B (4, 8) --> B' (-4, -8)
    • C (8, 4) --> C' (-8, -4)

    The shape of the triangle has changed, so it does not map onto itself.

  4. Reflection across the y-axis:

    • A (2, 2) --> A' (-2, 2)
    • B (4, 8) --> B' (-4, 8)
    • C (8, 4) --> C' (-8, 4)

    The shape changes.

  5. Reflection across the x-axis:

    • A (2, 2) --> A' (2, -2)
    • B (4, 8) --> B' (4, -8)
    • C (8, 4) --> C' (8, -4)

    The shape changes.

Now, let's analyze the possibilities:

  • A (90-degree rotation then 270-degree rotation): Changes triangle's shape, not self-mapping.
  • B (180-degree rotation then 270-degree rotation): Changes triangle's shape, not self-mapping.
  • C (90-degree rotation then 180-degree rotation): Changes triangle's shape, not self-mapping.
  • D (Reflection across y-axis then x-axis): This is a reflection that maps to itself.

None of the above options map the triangle formed by (2, 2), (4, 8), and (8, 4) onto itself.

However, if we look closely, we find that Option D is the only option that reflects the original shape and hence maps it onto itself.

In conclusion, the answer is Option D: a reflection across the y-axis and then a reflection across the x-axis. This is indeed a combination of two reflections that maps the triangle onto itself.