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The problem asks how to find the equation of a line that is perpendicular to the bisector of the line segment passing through the points \(\left( \frac{1}{2}, 10 \right)\) and \(\left( \frac{3}{2}, 4 \right)\).

Here’s how to approach the problem:

### Step 1: Find the midpoint of the line segment
The midpoint \( M \) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
For the points \(\left( \frac{1}{2}, 10 \right)\) and \(\left( \frac{3}{2}, 4 \right)\):
\[
M = \left( \frac{\frac{1}{2} + \frac{3}{2}}{2}, \frac{10 + 4}{2} \right) = \left( \frac{4}{4}, \frac{14}{2} \right) = (1, 7)
\]
So, the midpoint is \( M(1, 7) \).

### Step 2: Find the slope of the line segment
The slope \( m \) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
For the points \(\left( \frac{1}{2}, 10 \right)\) and \(\left( \frac{3}{2}, 4 \right)\):
\[
m = \frac{4 - 10}{\frac{3}{2} - \frac{1}{2}} = \frac{-6}{1} = -6
\]
So, the slope of the line segment is \( -6 \).

### Step 3: Find the slope of the perpendicular bisector
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Since the slope of the original line is \( -6 \), the slope of the perpendicular bisector is:
\[
m_{\perp} = \frac{1}{6}
\]

### Step 4: Write the equation of the perpendicular bisector
Now, we use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the midpoint \( M(1, 7) \) and the slope \( \frac{1}{6} \):
\[
y - 7 = \frac{1}{6}(x - 1)
\]
Simplifying:
\[
y - 7 = \frac{1}{6}x - \frac{1}{6}
\]
\[
y = \frac{1}{6}x - \frac{1}{6} + 7
\]
\[
y = \frac{1}{6}x + \frac{41}{6}
\]

So, the equation of the line that is perpendicular to the bisector of the segment is:
\[
y = \frac{1}{6}x + \frac{41}{6}
\]

Would you like a further breakdown or have any specific questions?

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Here are 5 questions related to this topic:
1. How do you find the equation of a line given two points?
2. What is the significance of a perpendicular bisector in geometry?
3. How do you calculate the midpoint of any line segment?
4. What is the point-slope form of a line equation, and when do you use it?
5. What is the relationship between the slopes of perpendicular lines?

**Tip:** Always remember that perpendicular lines have slopes that are negative reciprocals of each other!

¿Cómo encontraría una ecuación de la recta que es perpendicular a la bisectriz del segmento de recta que pasa por (1/2, 10) y (3/2, 4)?

Learn how to find the equation of a line that is perpendicular to the bisector of a segment connecting the points (1/2, 10) and (3/2, 4). This problem covers key concepts in coordinate geometry, including midpoint, slope calculation, and properties of perpendicular lines. Suitable for high school students in Grades 10-12.

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