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What is the equation of a parallel line through point ( 0 , -7) ?

To reflect the rhombus \( QRST \) over the line \( y = -2 \), we need to follow these steps for each vertex of the rhombus:

### Coordinates of the rhombus \( QRST \) vertices:
- \( Q(-6, 2) \)
- \( R(0, 0) \)
- \( S(2, 4) \)
- \( T(-4, 4) \)

The reflection formula for a point \((x, y)\) over the line \(y = -2\) is given by:
\[ y' = 2(-2) - y = -4 - y \]
So, for each vertex, the new \( y \)-coordinate will be found using this formula, while the \( x \)-coordinate remains the same.

### Reflection of each point:
1. **Point Q(-6, 2)**:
   - \( y' = -4 - 2 = -6 \)
   - New coordinates: \( Q'(-6, -6) \)

2. **Point R(0, 0)**:
   - \( y' = -4 - 0 = -4 \)
   - New coordinates: \( R'(0, -4) \)

3. **Point S(2, 4)**:
   - \( y' = -4 - 4 = -8 \)
   - New coordinates: \( S'(2, -8) \)

4. **Point T(-4, 4)**:
   - \( y' = -4 - 4 = -8 \)
   - New coordinates: \( T'(-4, -8) \)

### Reflected image \( Q'R'S'T' \):
The new vertices after reflection will be:
- \( Q'(-6, -6) \)
- \( R'(0, -4) \)
- \( S'(2, -8) \)
- \( T'(-4, -8) \)

Now for the equation of a line parallel to \( y = -2 \) passing through the point \( (0, -7) \), a parallel line will have the same slope, which is zero (since \( y = -2 \) is a horizontal line).

Thus, the equation of the parallel line passing through \( (0, -7) \) is:
\[
y = -7
\]

Let me know if you'd like more details or clarifications. Here are some related questions for further exploration:

1. How do you find the equation of a perpendicular line to \( y = -2 \) passing through \( (0, -7) \)?
2. What is the general formula for reflecting a point across any horizontal line \( y = k \)?
3. Can we determine the area of the rhombus before and after reflection?
4. How would you graphically represent the reflected rhombus on the same grid?
5. What changes in the properties of the rhombus after reflection?

**Tip:** When reflecting over a horizontal line, the \( x \)-coordinates remain the same, while only the \( y \)-coordinates are altered based on the reflection line.

Graph the image of rhombus QRST after a reflection over the line y = -2.

Learn how to reflect a rhombus QRST over the line y = -2. This problem covers key geometry concepts such as reflections, coordinate transformations, and includes the calculation of new vertex coordinates. Suitable for students in Grades 8-10.

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