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Chapter 3: Problem 93

Find an equation of the perpendicular bisector of the line segment whose endpoints are given. \((-2,6) ;(-22,-4)\)

Short Answer

Expert verified
The equation of the perpendicular bisector is \(y = -2x - 23\).

Step by step solution

01

Calculate the Midpoint

To find the equation of the perpendicular bisector, we first need the midpoint of the line segment with endpoints \((-2,6)\) and \((-22,-4)\). The formula for the midpoint \((x_m, y_m)\) is given by:\[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2} \]Substituting the given points into the formula, we get:\[ x_m = \frac{-2 + (-22)}{2} = \frac{-24}{2} = -12 \]\[ y_m = \frac{6 + (-4)}{2} = \frac{2}{2} = 1 \]Thus, the midpoint is \((-12, 1)\).
02

Determine the Slope of the Original Line

The slope \(m\) of the line segment connecting the points \((-2,6)\) and \((-22,-4)\) is found using the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the known values, we have:\[ m = \frac{-4 - 6}{-22 - (-2)} = \frac{-10}{-20} = \frac{1}{2} \]
03

Find the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the original line's slope. Since the original slope is \(\frac{1}{2}\), the perpendicular bisector's slope \(m_p\) is:\[ m_p = -\frac{1}{\frac{1}{2}} = -2 \]
04

Write the Equation of the Perpendicular Bisector

Now that we have the slope \(-2\) and the midpoint \((-12, 1)\), we can use the point-slope form of a line equation, which is:\[ y - y_1 = m(x - x_1) \]Substituting the midpoint and perpendicular slope into the equation:\[ y - 1 = -2(x + 12) \]Simplify to the slope-intercept form \(y = mx + b\):\[ y - 1 = -2x - 24 \]\[ y = -2x - 24 + 1 \]\[ y = -2x - 23 \]
05

Recheck the Calculations

Verify each calculation step to ensure accuracy:- Midpoint was determined correctly as \((-12,1)\).- Original slope \(\frac{1}{2}\) is correct.- Perpendicular slope \(-2\) is accurate as the negative reciprocal.- The equation \(y = -2x - 23\) fits the point \((-12,1)\) and has the proper slope.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Midpoint Calculation
Finding the midpoint of a line segment is like finding the middle point that splits the segment into two equal halves. If you have two endpoints, say (-2, 6) and (-22, -4), you apply the midpoint formula to find this central location.

The formula for the midpoint \((x_m, y_m)\) is:\[ x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}\]

Let's break it down with our points:
  • For the x-coordinate: \(-2\) and \(-22\) are averaged to find \(x_m\). So, \(x_m = \frac{-2 + (-22)}{2} = -12\).
  • For the y-coordinate: The points 6 and -4 are averaged to find \(y_m\). Thus, \(y_m = \frac{6 + (-4)}{2} = 1\).
The midpoint of our line segment is then \((-12, 1)\). This point will help us further in determining the perpendicular bisector.
Slope Determination
To understand how steep or flat our original line segment is, we calculate its slope, which tells us the rate of change in y with respect to x.

The general formula for slope \(m\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Using our given endpoints (-2,6) and (-22,-4), plug in the values:
  • The change in y is \(-4 - 6 = -10\).
  • The change in x is \(-22 - (-2) = -20\).
This gives us a slope \(m\) of:\[ m = \frac{-10}{-20} = \frac{1}{2} \]

Hence, the slope of the line segment is \(\frac{1}{2}\). A positive slope indicates the line rises as it moves from left to right.
Point-Slope Form
The point-slope form is a way of expressing the equation of a line using a known point and the slope. It's especially useful when exploring perpendicular bisectors.

The formula looks like this:\[ y - y_1 = m(x - x_1) \]

For our perpendicular bisector, we use:
  • The slope we've calculated, \(-2\),
  • Our midpoint \((-12, 1)\) as the point \((x_1, y_1)\).
Substituting these values, we have:\[ y - 1 = -2(x + 12) \]
Simplifying to find the y-intercept form gives:
  • Multiply: \(y - 1 = -2x - 24\).
  • Add 1: \(y = -2x - 23\).
This equation \(y = -2x - 23\) represents the perpendicular bisector, using the given midpoint and the slope.
Negative Reciprocal of Slope
A perpendicular bisector has a special relationship with the line segment it intersects. Its slope is the negative reciprocal of the original line's slope, flipping and changing its sign.

To find a negative reciprocal:
  • Take the given slope of the original line, \(\frac{1}{2}\).
  • Flip it: The reciprocal is \(\frac{2}{1}\).
  • Change the sign: The negative reciprocal becomes -2.
This new slope \(-2\) means the perpendicular bisector will slope in the opposite direction - a critical adjustment to ensure it bisects the segment at a right angle. This concept underscores how perpendicular structures relate in geometry, ensuring clarity and precision in plotting lines.

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