https://file.aiquickdraw.com/mathbot/file/1727249386177jpxw7bln.jpg

We are given the following problem:

- The diagram is part of a Voronoi diagram.
- A and B are sites, with \( B = (4, 6) \).
- The equation of the perpendicular bisector of the line segment from \( A \) to \( B \) is \( y = -2x + 9 \).

We are asked to find the coordinates of point \( A \).

### Step-by-step solution:

1. **Understanding the Perpendicular Bisector**:
   - The equation of the perpendicular bisector is \( y = -2x + 9 \). This bisector is the line that is equidistant from \( A \) and \( B \), meaning it passes through the midpoint of the line segment between \( A \) and \( B \), and is perpendicular to the line joining \( A \) and \( B \).

2. **Midpoint of \( A \) and \( B \)**:
   The perpendicular bisector passes through the midpoint of \( A(x_1, y_1) \) and \( B(4, 6) \). The formula for the midpoint is:
   \[
   \text{Midpoint} = \left( \frac{x_1 + 4}{2}, \frac{y_1 + 6}{2} \right)
   \]
   Since this point lies on the bisector \( y = -2x + 9 \), we will substitute this into the equation of the bisector once we have it.

3. **Slope of the Line Segment AB**:
   Let the coordinates of \( A \) be \( (x_1, y_1) \). The slope of the line connecting \( A(x_1, y_1) \) and \( B(4, 6) \) is:
   \[
   \text{Slope of AB} = \frac{y_1 - 6}{x_1 - 4}
   \]
   Since the line of the perpendicular bisector is perpendicular to \( AB \), the product of their slopes should equal \( -1 \). The slope of the bisector \( y = -2x + 9 \) is \( -2 \), so:
   \[
   \text{Slope of AB} \times (-2) = -1
   \]
   Solving for the slope of \( AB \):
   \[
   \frac{y_1 - 6}{x_1 - 4} = \frac{1}{2}
   \]

4. **Solving for \( y_1 \) in Terms of \( x_1 \)**:
   From the equation \( \frac{y_1 - 6}{x_1 - 4} = \frac{1}{2} \), we solve for \( y_1 \):
   \[
   y_1 - 6 = \frac{1}{2}(x_1 - 4)
   \]
   \[
   y_1 - 6 = \frac{1}{2}x_1 - 2
   \]
   \[
   y_1 = \frac{1}{2}x_1 + 4
   \]

5. **Substitute into the Midpoint Equation**:
   The midpoint is \( \left( \frac{x_1 + 4}{2}, \frac{\frac{1}{2}x_1 + 4 + 6}{2} \right) = \left( \frac{x_1 + 4}{2}, \frac{1}{2}x_1 + 5 \right) \).
   Since this point lies on the line \( y = -2x + 9 \), we substitute into the equation:
   \[
   \frac{1}{2}x_1 + 5 = -2\left( \frac{x_1 + 4}{2} \right) + 9
   \]
   Simplifying:
   \[
   \frac{1}{2}x_1 + 5 = -x_1 - 4 + 9
   \]
   \[
   \frac{1}{2}x_1 + 5 = -x_1 + 5
   \]
   Canceling the \( 5 \) terms:
   \[
   \frac{1}{2}x_1 = -x_1
   \]
   Multiplying both sides by 2:
   \[
   x_1 = -2x_1
   \]
   Solving for \( x_1 \):
   \[
   3x_1 = 0 \quad \Rightarrow \quad x_1 = 0
   \]

6. **Finding \( y_1 \)**:
   Substituting \( x_1 = 0 \) into \( y_1 = \frac{1}{2}x_1 + 4 \):
   \[
   y_1 = \frac{1}{2}(0) + 4 = 4
   \]

Thus, the coordinates of point \( A \) are \( \boxed{(0, 4)} \).

---

Would you like more details or clarifications?

Here are 5 related questions to deepen your understanding:
1. How do you derive the equation of a perpendicular bisector?
2. Why does the perpendicular bisector pass through the midpoint?
3. Can you find the coordinates of point \( A \) using a different method?
4. What is the significance of Voronoi diagrams in geometry?
5. How would the process change if the equation of the bisector had a different slope?

**Tip:** Always verify the solution by substituting the result back into the given conditions to ensure consistency.

Find the coordinates of the point A, given that B has coordinates (4, 6) and the equation for the perpendicular bisector of the line segment from A to B is y = -2x + 9.

Explore how to determine the coordinates of point A given point B at (4, 6) and the equation of the perpendicular bisector y = -2x + 9. This example uses geometry and coordinate geometry principles including midpoint and slope calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation