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Write the coordinates of the vertices after a reflection over the line y=1.


To find the coordinates of the vertices \( B \), \( C \), \( D \), and \( E \) after a reflection over the line \( y = 1 \), we need to follow these steps:

1. **Identify the coordinates of each vertex.**
   - \( B = (-8, -8) \)
   - \( C = (-7, -2) \)
   - \( D = (-4, 3) \)
   - \( E = (-5, -3) \)

2. **Reflect each point across the line \( y = 1 \).**
   - When reflecting a point \((x, y)\) over a horizontal line \(y = k\), the reflected point will be \((x, 2k - y)\).

Since \( y = 1 \), we can apply this formula to each vertex.

### Calculations

- **Point \( B = (-8, -8) \):**
  \[
  y' = 2(1) - (-8) = 2 + 8 = 10
  \]
  Reflected point of \( B \): \( (-8, 10) \).

- **Point \( C = (-7, -2) \):**
  \[
  y' = 2(1) - (-2) = 2 + 2 = 4
  \]
  Reflected point of \( C \): \( (-7, 4) \).

- **Point \( D = (-4, 3) \):**
  \[
  y' = 2(1) - 3 = 2 - 3 = -1
  \]
  Reflected point of \( D \): \( (-4, -1) \).

- **Point \( E = (-5, -3) \):**
  \[
  y' = 2(1) - (-3) = 2 + 3 = 5
  \]
  Reflected point of \( E \): \( (-5, 5) \).

### Reflected Coordinates

After reflecting over the line \( y = 1 \), the coordinates of the vertices are:

- \( B' = (-8, 10) \)
- \( C' = (-7, 4) \)
- \( D' = (-4, -1) \)
- \( E' = (-5, 5) \)

Would you like further details on how reflections work, or have any specific questions?

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Here are some follow-up questions:

1. How would the coordinates change if the reflection line was \( y = -1 \) instead of \( y = 1 \)?
2. What would be the new coordinates if the shape was rotated 90 degrees around the origin?
3. How can we generalize the reflection formula for any line \( y = k \)?
4. What if we wanted to reflect over a vertical line, say \( x = -4 \)?
5. How would you determine the distance between the original points and their reflected images?

**Tip:** For any point reflection, the distance between the original and the reflected point is always twice the distance from the original point to the line of reflection.

Write the coordinates of the vertices after a reflection over the line y=1.

Explore how to find the coordinates of vertices after reflecting over the line y=1. This problem involves key concepts in geometry, including reflections and transformations in coordinate geometry. Suitable for Grades 8-10.

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