find the equation of the line that is equidistant from the points (5,1) and (-4,-3). Write the equation in slope-intercept form

To find the equation of the line that is equidistant from the points \( (5, 1) \) and \( (-4, -3) \), we need to determine the perpendicular bisector of the segment connecting these two points. The steps are as follows:

### 1. Find the midpoint of the segment.
The midpoint of a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:

\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

For the points \( (5, 1) \) and \( (-4, -3) \):

\[
\text{Midpoint} = \left( \frac{5 + (-4)}{2}, \frac{1 + (-3)}{2} \right) = \left( \frac{1}{2}, \frac{-2}{2} \right) = \left( \frac{1}{2}, -1 \right)
\]

### 2. Find the slope of the line connecting the two points.
The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For \( (5, 1) \) and \( (-4, -3) \):

\[
m = \frac{-3 - 1}{-4 - 5} = \frac{-4}{-9} = \frac{4}{9}
\]

### 3. Find the slope of the perpendicular bisector.
The slope of a perpendicular line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular bisector is:

\[
m_{\perp} = -\frac{9}{4}
\]

### 4. Use the point-slope form to write the equation of the line.
The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Using the midpoint \( \left( \frac{1}{2}, -1 \right) \) and the perpendicular slope \( m_{\perp} = -\frac{9}{4} \):

\[
y - (-1) = -\frac{9}{4} \left( x - \frac{1}{2} \right)
\]

Simplify:

\[
y + 1 = -\frac{9}{4} \left( x - \frac{1}{2} \right)
\]

\[
y + 1 = -\frac{9}{4}x + \frac{9}{8}
\]

Subtract 1 from both sides:

\[
y = -\frac{9}{4}x + \frac{9}{8} - 1
\]

\[
y = -\frac{9}{4}x + \frac{9}{8} - \frac{8}{8}
\]

\[
y = -\frac{9}{4}x + \frac{1}{8}
\]

### Final Answer:
The equation of the line in slope-intercept form is:

\[
y = -\frac{9}{4}x + \frac{1}{8}
\]

Would you like further details on this, or any clarification?

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Here are 5 related questions to explore this topic further:

1. How do you find the equation of a line given two points?
2. What is the meaning of the perpendicular bisector in geometry?
3. How do you determine if two lines are perpendicular using their slopes?
4. What is the general form of the equation of a line?
5. How can you verify if a point lies on a given line?

**Tip:** When dealing with perpendicular lines, always remember that their slopes are negative reciprocals of each other!

Find the equation of the line that is equidistant from the points (5,1) and (-4,-3). Write the equation in slope-intercept form.

Find the equation of the perpendicular bisector that is equidistant from the points (5,1) and (-4,-3). This detailed solution walks through the steps of calculating the midpoint, determining slopes, and writing the final equation in slope-intercept form. Ideal for students in Grades 9-11.

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