The perpendicular bisector of the segment whose endpoints are parentheses -1, -3 parentheses and parentheses -3, four parentheses

To find the equation of the perpendicular bisector of a segment with endpoints \((-1, -3)\) and \((-3, 4)\), follow these steps:

### Step 1: Find the midpoint of the segment
The midpoint \(M\) of the segment is calculated using the midpoint formula:
\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]
Substituting the given points \((-1, -3)\) and \((-3, 4)\):
\[
M = \left(\frac{-1 + (-3)}{2}, \frac{-3 + 4}{2}\right) = \left(\frac{-4}{2}, \frac{1}{2}\right) = (-2, 0.5)
\]

### Step 2: Calculate the slope of the original segment
The slope \(m\) of the segment is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values:
\[
m = \frac{4 - (-3)}{-3 - (-1)} = \frac{7}{-2} = -\frac{7}{2}
\]

### Step 3: Determine the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the original slope. If the slope of the original segment is \(m\), then the slope \(m_{\text{bisector}}\) of the perpendicular bisector is:
\[
m_{\text{bisector}} = -\frac{1}{m} = -\frac{1}{-\frac{7}{2}} = \frac{2}{7}
\]

### Step 4: Write the equation of the perpendicular bisector
The equation of the line can be written in point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Using the midpoint \(M(-2, 0.5)\) and the slope \(\frac{2}{7}\):
\[
y - 0.5 = \frac{2}{7}(x + 2)
\]
Expanding and simplifying:
\[
y - 0.5 = \frac{2}{7}x + \frac{4}{7}
\]
Adding \(0.5\) (which is \(\frac{7}{14}\)) to both sides:
\[
y = \frac{2}{7}x + \frac{4}{7} + \frac{7}{14} = \frac{2}{7}x + \frac{4}{7} + \frac{2}{7} = \frac{2}{7}x + \frac{6}{7}
\]
So, the equation of the perpendicular bisector is:
\[
y = \frac{2}{7}x + \frac{6}{7}
\]

Would you like to go over any details or have any questions?

Here are some related questions you might find useful:
1. How do you find the distance between two points?
2. What is the significance of the negative reciprocal when finding perpendicular slopes?
3. Can you derive the perpendicular bisector equation if the points were swapped?
4. How can the midpoint formula be applied in real-life situations?
5. How do you check if a point lies on a given line?

**Tip:** When solving problems involving perpendicular lines, always remember that their slopes are negative reciprocals of each other.

Learn how to find the equation of the perpendicular bisector for the segment with endpoints (-1, -3) and (-3, 4). This problem involves calculating the midpoint, determining slopes, and applying the point-slope form of a line. Suitable for high school students studying coordinate geometry.

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