To determine the vertices of square A′B′C′D′ after reflecting square ABCD first across the x-axis and then across the y-axis, let's consider the process step by step.
Assuming that the vertices of square ABCD are as follows:
- A(x₁, y₁)
- B(x₂, y₂)
- C(x₃, y₃)
- D(x₄, y₄)
Step 1: Reflect across the x-axis
When reflecting a point (x, y) across the x-axis, the new coordinates become (x, -y).
Thus, the vertices after reflecting across the x-axis will be:
- A′(x₁, -y₁)
- B′(x₂, -y₂)
- C′(x₃, -y₃)
- D′(x₄, -y₄)
Step 2: Reflect across the y-axis
Next, we reflect the result across the y-axis. When reflecting a point (x, y) across the y-axis, the new coordinates become (-x, y).
Therefore, the vertices after reflecting across the y-axis will become:
- A′′(−x₁, -y₁)
- B′′(−x₂, -y₂)
- C′′(−x₃, -y₃)
- D′′(−x₄, -y₄)
Now, let's identify the original coordinates for square ABCD.
Assume the original square ABCD has the following coordinates:
- A(3, 4)
- B(7, 4)
- C(7, 8)
- D(3, 8)
Apply Step 1 (reflection across the x-axis):
- A′(3, -4)
- B′(7, -4)
- C′(7, -8)
- D′(3, -8)
Apply Step 2 (reflection across the y-axis):
- A′′(−3, -4)
- B′′(−7, -4)
- C′′(−7, -8)
- D′′(−3, -8)
Therefore, the vertices of square A′B′C′D′ after both reflections are: A′(−3, −4), B′(−7, −4), C′(−7, −8), and D′(−3, −8).
Hence, the correct response is: Square A′B′C′D′ has vertices A′(−3, −4), B′(−7, −4), C′(−7, −8), and D′(−3, −8).
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