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We are tasked with finding the coordinates of point A, given the following information:

- Point B has coordinates \( (4, 6) \).
- The equation of the perpendicular bisector for the line segment joining A and B is given as \( y = -2x + 9 \).
- Line \( L \) is part of a Voronoi diagram, and it is the perpendicular bisector of the line segment between A and B.

### Steps:

1. **Midpoint of A and B**:  
   Since the perpendicular bisector passes through the midpoint of \( A \) and \( B \), we can first find the midpoint of the line segment from \( A(x_1, y_1) \) to \( B(4, 6) \). The midpoint \( M \) is given by:
   \[
   M = \left( \frac{x_1 + 4}{2}, \frac{y_1 + 6}{2} \right)
   \]

2. **Perpendicular Slope Relation**:  
   The slope of the line segment from \( A \) to \( B \) will be perpendicular to the slope of the bisector \( y = -2x + 9 \). Since the slope of the bisector is \( -2 \), the slope of the line connecting A and B is the negative reciprocal, \( \frac{1}{2} \).

3. **Equation of Line AB**:  
   Using the slope \( \frac{1}{2} \) and the point \( B(4, 6) \), we can write the equation of the line \( AB \). Using the point-slope form \( y - y_1 = m(x - x_1) \), we substitute \( (4, 6) \) and slope \( \frac{1}{2} \):
   \[
   y - 6 = \frac{1}{2}(x - 4)
   \]

4. **Solving for A**:  
   The coordinates of A can be determined by solving this system involving the line equation of \( AB \) and the midpoint.

Let me compute that.The coordinates of point \( A \) are \( (0, 4) \).

Would you like further explanation on how these were calculated, or any additional details?

Here are 5 related questions to explore further:
1. How do we derive the equation of a perpendicular bisector given two points?
2. Why is the slope of the perpendicular bisector the negative reciprocal of the original line's slope?
3. How does this problem relate to Voronoi diagrams?
4. How would the solution change if the coordinates of point B were different?
5. Can this method be used for three-dimensional points?

**Tip**: Always remember, the midpoint lies on the perpendicular bisector, which is a crucial fact when solving these types of problems.

The diagram below is part of a Voronoi diagram. L is an edge. A and B are sites with B having the coordinates of (4, 6). The equation for the perpendicular bisector for the line segment from A to B is y = -2x + 9. Find the coordinates of the point A.

Solve for the coordinates of point A in a Voronoi diagram where B is located at (4, 6) and the equation for the perpendicular bisector is y = -2x + 9. Learn key concepts in coordinate geometry including the midpoint formula and perpendicular bisector properties.

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