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Reflect square ABCD first across the z-axis, and then across the y-axis to create square A' * B' * C' * D' What are the vertices of square A' * B' * C' * D' ?

(1 point)

Square A'B'C'D' has vertices A'(- 3, 4) B (-7, 4), and C' (- 7, 8).

Square A'B'C'D' has vertices A'(3, 4) B (7,4), C'(7, 8) and D'(3, 8).

Square A'B'C'D' has vertices A'(- 3, - 4) B'(- 7, - 4) C'(- 7, - 8) and D'(- 3, - 8).

Square A'B'C'D' has vertices A'(- 3, 4) B'(- 7, 4) C'(- 7, 8) and D'(- 3, 8).

1 answer

To find the vertices of square A'B'C'D' after reflecting square ABCD across the z-axis and then the y-axis, let's begin by determining the transformations for these reflections.

  1. Reflection across the z-axis: This reflection changes the sign of the x-coordinates of the points. If the original points are (x, y), then after reflecting across the z-axis, they become (-x, y).

  2. Reflection across the y-axis: This reflection changes the sign of the y-coordinates. After reflecting (-x, y) across the y-axis, we get (-x, -y).

Now, starting from square ABCD, let's denote its vertices as follows:

  • A(x1, y1)
  • B(x2, y2)
  • C(x3, y3)
  • D(x4, y4)

After the first transformation (reflection across the z-axis):

  • A' = (-x1, y1)
  • B' = (-x2, y2)
  • C' = (-x3, y3)
  • D' = (-x4, y4)

After the second transformation (reflection across the y-axis), we get:

  • A'' = (-(-x1), -y1) = (x1, -y1)
  • B'' = (-(-x2), -y2) = (x2, -y2)
  • C'' = (-(-x3), -y3) = (x3, -y3)
  • D'' = (-(-x4), -y4) = (x4, -y4)

Now, we need the original coordinates of A, B, C, and D. Unfortunately, these were not provided, but let's evaluate based on the choices systematically.

According to the options given for square A'B'C'D':

  • A'(-3, 4), B'(-7, 4), C'(-7, 8), D'(-3, 8)
  • A'(3, 4), B(7,4), C'(7, 8), D'(3, 8)
  • A'(-3, -4), B'(-7, -4), C'(-7, -8), D'(-3, -8)
  • A'(-3, 4), B'(-7, 4), C'(-7, 8), D'(-3, 8)

We have two transformations - reflection over the z-axis and y-axis, and we need to consider their combined effect.

  • If we take the first choice, A'(-3, 4), B'(-7, 4), C'(-7, 8), and D'(-3, 8), this means the original square needed to be situated in the first quadrant, as the y-coordinates are positive.
  • The one choice (3, 4) etc. indicates a square in the positive coordinates.

We can conclude that the coordinates have either the reflection impacting them negatively, hence looking at the third option (A'(-3, -4), B'(-7, -4), C'(-7, -8), D'(-3, -8)) indicates they result in a correctly transformed point from the assumed squares based on the rules we went through.

The correct answer is: Square A'B'C'D' has vertices A'(-3, -4), B'(-7, -4), C'(-7, -8), and D'(-3, -8).