To find the coordinates of the vertices of rectangle ABCD after it is reflected on the x-axis and then on the y-axis, we will apply the reflection transformations to each vertex of the rectangle.
Step 1: Reflect over the x-axis. When a point \((x, y)\) is reflected over the x-axis, its new coordinates are \((x, -y)\).
Applying this to each vertex:
- A (-1, -2) → A' (-1, 2)
- B (-1, -8) → B' (-1, 8)
- C (-8, -8) → C' (-8, 8)
- D (-8, -2) → D' (-8, 2)
Step 2: Reflect over the y-axis. When a point \((x, y)\) is reflected over the y-axis, its new coordinates are \((-x, y)\).
Now, we apply this transformation to the vertices obtained after the first reflection:
- A' (-1, 2) → A'' (1, 2)
- B' (-1, 8) → B'' (1, 8)
- C' (-8, 8) → C'' (8, 8)
- D' (-8, 2) → D'' (8, 2)
Thus, the coordinates of the vertices for rectangle A''B''C''D'' after reflecting rectangle ABCD over the x-axis and then the y-axis are:
- A'' (1, 2)
- B'' (1, 8)
- C'' (8, 8)
- D'' (8, 2)
So, the final coordinates of the vertices are:
- A'' (1, 2)
- B'' (1, 8)
- C'' (8, 8)
- D'' (8, 2)
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