Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![**Equation of a Perpendicular Bisector**
**Objective:** Find an equation for the perpendicular bisector of the line segment whose endpoints are (4, -3) and (-1, 7).
To solve this problem, follow these steps:
1. **Determine the midpoint of the segment:**
- The midpoint \(M\) of a line segment whose endpoints are \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
- Substitute the given points \((4, -3)\) and \((-1, 7)\) into the formula:
\[
M = \left( \frac{4 + (-1)}{2}, \frac{-3 + 7}{2} \right) = \left( \frac{3}{2}, 2 \right)
\]
2. **Calculate the slope of the line segment:**
- The slope \(m\) of the line through \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
- Substitute the given points:
\[
m = \frac{7 - (-3)}{-1 - 4} = \frac{10}{-5} = -2
\]
3. **Find the slope of the perpendicular bisector:**
- The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line segment.
\[
m_{\text{perp}} = -\frac{1}{m} = -\frac{1}{-2} = \frac{1}{2}
\]
4. **Write the equation of the perpendicular bisector:**
- Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line (in this case, the midpoint).
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Transcribed Image Text:**Equation of a Perpendicular Bisector**
**Objective:** Find an equation for the perpendicular bisector of the line segment whose endpoints are (4, -3) and (-1, 7).
To solve this problem, follow these steps:
1. **Determine the midpoint of the segment:**
- The midpoint \(M\) of a line segment whose endpoints are \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
- Substitute the given points \((4, -3)\) and \((-1, 7)\) into the formula:
\[
M = \left( \frac{4 + (-1)}{2}, \frac{-3 + 7}{2} \right) = \left( \frac{3}{2}, 2 \right)
\]
2. **Calculate the slope of the line segment:**
- The slope \(m\) of the line through \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
- Substitute the given points:
\[
m = \frac{7 - (-3)}{-1 - 4} = \frac{10}{-5} = -2
\]
3. **Find the slope of the perpendicular bisector:**
- The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line segment.
\[
m_{\text{perp}} = -\frac{1}{m} = -\frac{1}{-2} = \frac{1}{2}
\]
4. **Write the equation of the perpendicular bisector:**
- Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line (in this case, the midpoint).
\[
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