Ask a New Question
Menu

A triangle has vertices at B(−3, 0), C(2, −1), D(−1, 2). Which transformation would produce an image with vertices B″(−2, 1), C″(3, 2), D″(0, −1)?

(x, y) → (x, −y) → (x + 1, y + 1)
(x, y) → (−x, y) → (x + 1, y + 1)
(x, y) → (x, −y) → (x + 2, y + 2)
(x, y) → (−x, y) → (x + 2, y + 2)

1 answer

To find which transformation produces the specified image from the original triangle, we need to examine each step in the proposed transformations and check if they yield the correct vertices for B″(−2, 1), C″(3, 2), and D″(0, −1).

The original vertices are:

  • B(−3, 0),
  • C(2, −1),
  • D(−1, 2).

Now let's apply each provided transformation step-by-step and see if we get the new vertices.

  1. Transformation 1: (x, y) → (x, −y) → (x + 1, y + 1)

    • First step: (x, y) → (x, −y)

      • B(−3, 0) → B'(-3, 0)
      • C(2, −1) → C'(2, 1)
      • D(−1, 2) → D'(-1, -2)

    • Second step: (x, y) → (x + 1, y + 1)

      • B'(-3, 0) → B''(-3 + 1, 0 + 1) = B''(-2, 1)
      • C'(2, 1) → C''(2 + 1, 1 + 1) = C''(3, 2)
      • D'(-1, -2) → D''(-1 + 1, -2 + 1) = D''(0, -1)

    This transformation gives us B''(−2, 1), C''(3, 2), D''(0, −1).

  2. Transformation 2: (x, y) → (−x, y) → (x + 1, y + 1)

    • First step: (x, y) → (−x, y)

      • B(−3, 0) → B'(3, 0)
      • C(2, −1) → C'(-2, -1)
      • D(−1, 2) → D'(1, 2)

    • Second step: (x, y) → (x + 1, y + 1)

      • B'(3, 0) → B''(3 + 1, 0 + 1) = B''(4, 1)
      • C'(-2, -1) → C''(-2 + 1, -1 + 1) = C''(-1, 0)
      • D'(1, 2) → D''(1 + 1, 2 + 1) = D''(2, 3)

    This transformation does not yield the correct new vertices.

  3. Transformation 3: (x, y) → (x, −y) → (x + 2, y + 2)

    • First step: (x, y) → (x, −y)

      • B(−3, 0) → B'(-3, 0)
      • C(2, −1) → C'(2, 1)
      • D(−1, 2) → D'(-1, -2)

    • Second step: (x, y) → (x + 2, y + 2)

      • B'(-3, 0) → B''(-3 + 2, 0 + 2) = B''(-1, 2)
      • C'(2, 1) → C''(2 + 2, 1 + 2) = C''(4, 3)
      • D'(-1, -2) → D''(-1 + 2, -2 + 2) = D''(1, 0)

    This transformation does not yield the correct new vertices.

  4. Transformation 4: (x, y) → (−x, y) → (x + 2, y + 2)

    • First step: (x, y) → (−x, y)

      • B(−3, 0) → B'(3, 0)
      • C(2, −1) → C'(-2, -1)
      • D(−1, 2) → D'(1, 2)

    • Second step: (x, y) → (x + 2, y + 2)

      • B'(3, 0) → B''(3 + 2, 0 + 2) = B''(5, 2)
      • C'(-2, -1) → C''(-2 + 2, -1 + 2) = C''(0, 1)
      • D'(1, 2) → D''(1 + 2, 2 + 2) = D''(3, 4)

    This transformation does not yield the correct new vertices.

Thus, the only transformation that leads to B″(−2, 1), C″(3, 2), and D″(0, −1) is the first transformation:

(x, y) → (x, −y) → (x + 1, y + 1).