Question

A line segment has endpoints at (–1, 4) and (4, 1). Which reflection will produce an image with endpoints at (–4, 1) and (–1, –4)?

a reflection of the line segment across the x-axis
a reflection of the line segment across the y-axis
a reflection of the line segment across the line y = x
a reflection of the line segment across the line y = –x

Answer 1

To determine which reflection produces the desired image for the endpoints of the line segment, we start by examining the endpoints of the original line segment and the desired image:

- Original endpoints: \((-1, 4)\) and \((4, 1)\)
- Desired endpoints: \((-4, 1)\) and \((-1, -4)\)

Now, let's explore each reflection option:

1. **Reflection across the x-axis**:
- \( (x, y) \) becomes \( (x, -y) \)
- Reflecting \((-1, 4)\) gives \((-1, -4)\)
- Reflecting \((4, 1)\) gives \((4, -1)\)
- Resulting points: \((-1, -4)\) and \( (4, -1)\) (not what we need)

2. **Reflection across the y-axis**:
- \( (x, y) \) becomes \( (-x, y) \)
- Reflecting \((-1, 4)\) gives \((1, 4)\)
- Reflecting \((4, 1)\) gives \((-4, 1)\)
- Resulting points: \((1, 4)\) and \((-4, 1)\) (not what we need)

3. **Reflection across the line \(y = x\)**:
- \( (x, y) \) becomes \( (y, x) \)
- Reflecting \((-1, 4)\) gives \((4, -1)\)
- Reflecting \((4, 1)\) gives \((1, 4)\)
- Resulting points: \((4, -1)\) and \((1, 4)\) (not what we need)

4. **Reflection across the line \(y = -x\)**:
- \( (x, y) \) becomes \((-y, -x)\)
- Reflecting \((-1, 4)\) gives \((-4, 1)\)
- Reflecting \((4, 1)\) gives \((-1, -4)\)
- Resulting points: \((-4, 1)\) and \((-1, -4)\) (which matches our desired endpoints)

Based on this analysis, the correct reflection is **a reflection of the line segment across the line \(y = -x\)**.