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Magazine Chapter 10: Problem 23
Find an equation of the line described. Leave the solution in the form \(A x+B y=C\). The line is the perpendicular bisector of the line segment that joins \((3,5)\) and \((5,-1)\)
Short Answer
Expert verified
The equation is \(x - 3y = -2\).
Step by step solution
01
Find the midpoint of the line segment
The line segment joins the points \((3, 5)\) and \((5, -1)\). To find the midpoint, use the midpoint formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).\]Substituting in the given points:\[\left( \frac{3 + 5}{2}, \frac{5 + (-1)}{2} \right) = (4, 2). \]Thus, the midpoint of the line segment is \((4, 2)\).
02
Calculate the slope of the line segment
The slope \(m\) of the line segment connecting \((3, 5)\) and \((5, -1)\) is given by the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 5}{5 - 3} = \frac{-6}{2} = -3.\]
03
Find the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of the segment. If the original slope is \(m = -3\), the perpendicular slope \(m_\perp\) is:\[m_\perp = -\frac{1}{-3} = \frac{1}{3}.\]
04
Write the equation of the perpendicular bisector in point-slope form
Using the point \((4, 2)\) from Step 1 and the slope \(\frac{1}{3}\) from Step 3, apply the point-slope formula:\[y - y_1 = m(x - x_1).\]Substitute \((x_1, y_1) = (4, 2)\) and \(m = \frac{1}{3}\) to get:\[y - 2 = \frac{1}{3}(x - 4).\]
05
Convert the equation to the form \(Ax + By = C\)
Starting with the equation from Step 4:\[y - 2 = \frac{1}{3}(x - 4).\]First, clear the fraction by multiplying every term by 3:\[3(y - 2) = x - 4.\]Simplify:\[3y - 6 = x - 4.\]Rearrange terms to obtain:\[x - 3y = -2.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint Formula
The midpoint formula is a helpful tool when dealing with line segments. To find the midpoint between two endpoints, you need to calculate the average of the x-coordinates and the average of the y-coordinates.
Here's the formula: - **Midpoint (M)** is given by \ \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \]Let's break down the process with an example using the points \((3, 5)\) and \((5, -1)\):
Here's the formula: - **Midpoint (M)** is given by \ \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \]Let's break down the process with an example using the points \((3, 5)\) and \((5, -1)\):
- Add the x-coordinates: \(3 + 5 = 8\)
- Divide by 2 to find the x-coordinate of the midpoint: \(\frac{8}{2} = 4\)
- Add the y-coordinates: \(5 + (-1) = 4\)
- Divide by 2 to find the y-coordinate of the midpoint: \(\frac{4}{2} = 2\)
Slope of a Line
Understanding the slope of a line is crucial in determining how steep a line is and the direction it takes. The slope is a measure that describes the change in the y-coordinate as the x-coordinate changes.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: - **Slope formula**: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Consider the line segment with points \((3, 5)\) and \((5, -1)\). To find its slope:
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: - **Slope formula**: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Consider the line segment with points \((3, 5)\) and \((5, -1)\). To find its slope:
- Subtract the y-coordinates: \(-1 - 5 = -6\)
- Subtract the x-coordinates: \(5 - 3 = 2\)
- Calculate the slope: \(\frac{-6}{2} = -3\)
Negative Reciprocal
The negative reciprocal plays a key role when working with perpendicular lines. Given a line with a slope \(m\), a line that is perpendicular to it will have a slope that is the negative reciprocal of \(m\).
Here's how it works: - If the original slope is \(m\), then the perpendicular slope \(m_\perp\) is expressed as: \[ m_\perp = -\frac{1}{m} \]For example, if we have a line with slope \(-3\), the perpendicular slope would be:
Here's how it works: - If the original slope is \(m\), then the perpendicular slope \(m_\perp\) is expressed as: \[ m_\perp = -\frac{1}{m} \]For example, if we have a line with slope \(-3\), the perpendicular slope would be:
- Calculate the negative reciprocal: \[-\frac{1}{-3} = \frac{1}{3}\]
Point-Slope Form
The point-slope form of a line's equation is very useful when you know a point on the line and the line's slope. The point-slope form is helpful for writing equations of lines quickly and easily.
The general form is: - **Point-slope form**: \[ y - y_1 = m(x - x_1) \]Where \((x_1, y_1)\) is a specific point on the line and \(m\) is the slope. Consider this example: We have a midpoint \((4, 2)\) and a slope of \(\frac{1}{3}\). Using the formula:
The general form is: - **Point-slope form**: \[ y - y_1 = m(x - x_1) \]Where \((x_1, y_1)\) is a specific point on the line and \(m\) is the slope. Consider this example: We have a midpoint \((4, 2)\) and a slope of \(\frac{1}{3}\). Using the formula:
- Substitute \((x_1, y_1) = (4, 2)\) and \(m = \frac{1}{3}\):\[y - 2 = \frac{1}{3}(x - 4)\]
