Question

Which statement accurately explains whether a reflection over the x-axis and a 180° rotation would map figure ACB onto itself?
a coordinate plane with figure ACB with point A at 1, 1, C at 3, 4 and B at 5, 1
No, A″C″B″ is located at A″(−1, 1), C″(−3, 4), and B″(−5, 1)
No, A″C″B″ is located at A″(1, 1), C″(3, 4), and B″(5, 1)
Yes, A″C″B″ is located at A″(−1, 1), C″(−3, 4), and B″(−5, 1)
Yes, A″C″B″ is located at A″(1, 1), C″(3, 4), and B″(5, 1)

Answer 1

To determine whether a reflection over the x-axis and a 180° rotation would map figure ACB onto itself, we need to analyze the transformations.

1. **Reflection over the x-axis**:
- When a point \((x, y)\) is reflected over the x-axis, its new coordinates become \((x, -y)\).
- For the points A (1, 1), C (3, 4), and B (5, 1):
- A becomes A' (1, -1)
- C becomes C' (3, -4)
- B becomes B' (5, -1)

2. **180° Rotation**:
- A 180° rotation about the origin transforms a point \((x, y)\) to \((-x, -y)\).
- Applying this to the reflected points from above:
- A' (1, -1) becomes A'' (-1, 1)
- C' (3, -4) becomes C'' (-3, 4)
- B' (5, -1) becomes B'' (-5, 1)

After both transformations, we have:
- A'' = (-1, 1)
- C'' = (-3, 4)
- B'' = (-5, 1)

Now, we compare these new coordinates with the original coordinates:
- A (1, 1), C (3, 4), B (5, 1)

This shows that ACB does not map onto itself after these transformations.

Thus, the correct statement is:
**No, A″C″B″ is located at A″(−1, 1), C″(−3, 4), and B″(−5, 1).**