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
Related questions
Question
![**Problem:**
Find an equation for the perpendicular bisector of the line segment whose endpoints are (3, -1) and (-1, 3).
**Solution:**
To solve this problem, we need to follow these steps:
1. **Find the midpoint of the line segment:**
The formula for the midpoint, \( M \), of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Given the endpoints \((3, -1)\) and \((-1, 3)\):
\[
M = \left( \frac{3 + (-1)}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1)
\]
2. **Find the slope of the line segment:**
The formula for the slope, \( m \), of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Given the endpoints \((3, -1)\) and \((-1, 3)\):
\[
m = \frac{3 - (-1)}{-1 - 3} = \frac{4}{-4} = -1
\]
3. **Determine the slope of the perpendicular bisector:**
The slope of the perpendicular bisector of a line segment is the negative reciprocal of the slope of the line segment. If the slope of the line segment is \( m \), then the slope of the perpendicular bisector is \( -\frac{1}{m} \).
Given the slope of the line segment is \( -1 \):
\[
\text{slope of the perpendicular bisector} = -\frac{1}{-1} = 1
\]
4. **Find the equation of the perpendicular bisector:**
The equation of a line with](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9500080-40a0-4cd0-8ef1-168bcceb21f9%2F6cd4b7f8-46e5-4fcd-ac7f-279ba37cf3a0%2F22lojts_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find an equation for the perpendicular bisector of the line segment whose endpoints are (3, -1) and (-1, 3).
**Solution:**
To solve this problem, we need to follow these steps:
1. **Find the midpoint of the line segment:**
The formula for the midpoint, \( M \), of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Given the endpoints \((3, -1)\) and \((-1, 3)\):
\[
M = \left( \frac{3 + (-1)}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1)
\]
2. **Find the slope of the line segment:**
The formula for the slope, \( m \), of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Given the endpoints \((3, -1)\) and \((-1, 3)\):
\[
m = \frac{3 - (-1)}{-1 - 3} = \frac{4}{-4} = -1
\]
3. **Determine the slope of the perpendicular bisector:**
The slope of the perpendicular bisector of a line segment is the negative reciprocal of the slope of the line segment. If the slope of the line segment is \( m \), then the slope of the perpendicular bisector is \( -\frac{1}{m} \).
Given the slope of the line segment is \( -1 \):
\[
\text{slope of the perpendicular bisector} = -\frac{1}{-1} = 1
\]
4. **Find the equation of the perpendicular bisector:**
The equation of a line with
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