The image of trapezoid ABCD has coordinates A′(–2, –5), B′(–1, –2), C′(2, –2), and D′(3, –5). It was translated by the rule T–1, –3(x, y). Which diagram shows the pre-image?

On a coordinate plane, a trapezoid has points A (negative 1, negative 2), B (0, 1), C (3, 1), D (4, negative 2).

On a coordinate plane, a trapezoid has points A (negative 3, negative 2), B (negative 2, 1), C (1, 1), D (2, negative 2).

On a coordinate plane, a trapezoid has points A (negative 4, negative 5), B (negative 3, negative 2), C (0, negative 2), D (1, negative 5).

On a coordinate plane, a trapezoid has points A (0, negative 2), B (1, 1), C (4, 1), D (5, negative 2).

1 answer

To find the pre-image coordinates of trapezoid ABCD after applying the translation rule \( T_{-1, -3}(x, y) \), we need to reverse the translation. The translation rule \( T_{-1, -3}(x, y) \) means that every point (x, y) in the pre-image is transformed to (x - 1, y - 3) in the image.

Therefore, to find the pre-image coordinates A, B, C, and D from the image coordinates A′, B′, C′, and D′, we can apply the following transformations:

  • For point A′(–2, –5): \[ A(-1, -2) = A'(-2 + 1, -5 + 3) = (-1, -2) \]

  • For point B′(–1, –2): \[ B(0, 1) = B'(-1 + 1, -2 + 3) = (0, 1) \]

  • For point C′(2, –2): \[ C(3, 1) = C'(2 + 1, -2 + 3) = (3, 1) \]

  • For point D′(3, –5): \[ D(4, -2) = D'(3 + 1, -5 + 3) = (4, -2) \]

Thus, the pre-image coordinates are:

  • A(–1, –2)
  • B(0, 1)
  • C(3, 1)
  • D(4, –2)

Now, let’s check which of the given trapezoids corresponds to these points:

  1. First option: A(–1, –2), B(0, 1), C(3, 1), D(4, –2) — This matches our pre-image.

  2. Second option: A(–3, –2), B(–2, 1), C(1, 1), D(2, –2) — This does not match.

  3. Third option: A(–4, –5), B(–3, –2), C(0, –2), D(1, –5) — This does not match.

  4. Fourth option: A(0, –2), B(1, 1), C(4, 1), D(5, –2) — This does not match.

Therefore, the correct diagram that shows the pre-image of trapezoid ABCD is the first option:

  • A(–1, –2), B(0, 1), C(3, 1), D(4, –2).