Determine an equation for the perpendicular bisector of a line segment with each pair of endpoints. a) C(—2, 0) and D(4, —4) c) L(—2, —4) and M(8, 4)

To determine the equation of the perpendicular bisector of a line segment, we follow these steps:

1. **Find the midpoint** of the line segment, which will lie on the perpendicular bisector.
2. **Determine the slope** of the line segment, and then find the negative reciprocal of this slope (because the bisector is perpendicular).
3. **Use the point-slope form** of a line equation to derive the equation of the perpendicular bisector.

### Part (a): C(-2, 0) and D(4, -4)

1. **Midpoint of CD**:
   The midpoint \( M \) of the line segment with endpoints \( (x_1, y_1) = (-2, 0) \) and \( (x_2, y_2) = (4, -4) \) is given by:
   \[
   M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-2 + 4}{2}, \frac{0 - 4}{2} \right) = (1, -2)
   \]

2. **Slope of CD**:
   The slope \( m \) of the line segment CD is:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}
   \]

3. **Slope of the perpendicular bisector**:
   The slope of the perpendicular bisector is the negative reciprocal of \( -\frac{2}{3} \), which is \( \frac{3}{2} \).

4. **Equation of the perpendicular bisector**:
   Using the point-slope form \( y - y_1 = m(x - x_1) \) with the midpoint \( M(1, -2) \) and the slope \( \frac{3}{2} \):
   \[
   y + 2 = \frac{3}{2}(x - 1)
   \]
   Distribute the \( \frac{3}{2} \):
   \[
   y + 2 = \frac{3}{2}x - \frac{3}{2}
   \]
   Subtract 2 from both sides:
   \[
   y = \frac{3}{2}x - \frac{7}{2}
   \]
   So, the equation of the perpendicular bisector is:
   \[
   y = \frac{3}{2}x - \frac{7}{2}
   \]

### Part (c): L(-2, -4) and M(8, 4)

1. **Midpoint of LM**:
   The midpoint \( N \) of the line segment with endpoints \( (x_1, y_1) = (-2, -4) \) and \( (x_2, y_2) = (8, 4) \) is given by:
   \[
   N = \left( \frac{-2 + 8}{2}, \frac{-4 + 4}{2} \right) = (3, 0)
   \]

2. **Slope of LM**:
   The slope \( m \) of the line segment LM is:
   \[
   m = \frac{4 - (-4)}{8 - (-2)} = \frac{8}{10} = \frac{4}{5}
   \]

3. **Slope of the perpendicular bisector**:
   The slope of the perpendicular bisector is the negative reciprocal of \( \frac{4}{5} \), which is \( -\frac{5}{4} \).

4. **Equation of the perpendicular bisector**:
   Using the point-slope form with the midpoint \( N(3, 0) \) and the slope \( -\frac{5}{4} \):
   \[
   y - 0 = -\frac{5}{4}(x - 3)
   \]
   Distribute the \( -\frac{5}{4} \):
   \[
   y = -\frac{5}{4}x + \frac{15}{4}
   \]
   So, the equation of the perpendicular bisector is:
   \[
   y = -\frac{5}{4}x + \frac{15}{4}
   \]

### Summary of Results:
- (a) Perpendicular bisector: \( y = \frac{3}{2}x - \frac{7}{2} \)
- (c) Perpendicular bisector: \( y = -\frac{5}{4}x + \frac{15}{4} \)

Would you like further details on any step, or clarification on how these results were obtained?

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Here are five related questions to explore further:

1. How do you determine the equation of a line given two points?
2. What is the significance of a perpendicular bisector in geometry?
3. How can you find the intersection point of two lines?
4. What are some real-life applications of perpendicular bisectors?
5. How do you convert a line equation from point-slope form to slope-intercept form?

**Tip:** When finding slopes of perpendicular lines, remember that their slopes are negative reciprocals of each other.

Determine an equation for the perpendicular bisector of a line segment with each pair of endpoints. a) C(-2, 0) and D(4, -4) c) L(-2, -4) and M(8, 4)

Learn how to find the equation of the perpendicular bisector of a line segment using the points C(-2, 0) and D(4, -4), and L(-2, -4) and M(8, 4). This solution covers midpoint and slope formulas, and demonstrates step-by-step how to derive the equation of a line. Suitable for students in Grades 9-11.

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