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Magazine Chapter 1: Problem 56
Find the equation of the perpendicular bisector of the line segment joining the two given points. $$(-6,2),(6,-7)$$
Short Answer
Expert verified
Answer: The equation of the perpendicular bisector is 8x - 6y = 15.
Step by step solution
01
Find the midpoint of the line segment joining the given points
To find the midpoint \((x_m, y_m)\) of the line segment joining the points \((-6, 2)\) and \((6, -7)\), we can use the midpoint formula:
$$x_m = \frac{x_1+x_2}{2}$$
$$y_m = \frac{y_1+y_2}{2}$$
So, in our case, \(x_m = \frac{-6+6}{2}\) and \(y_m = \frac{2-7}{2}\).
02
Calculate the midpoint coordinates
Computing the midpoint coordinates, we get:
$$x_m = \frac{-6+6}{2}=0$$
$$y_m = \frac{2-7}{2}=-\frac{5}{2}$$
Thus, the midpoint of the line segment is \((0, -\frac{5}{2})\).
03
Determine the slope of the original line
To find the slope \(m\) of the line containing points \((-6, 2)\) and \((6, -7)\), we can use the slope formula:
$$m = \frac{y_2-y_1}{x_2-x_1}$$
Substituting the given coordinates, we get:
$$m = \frac{-7-2}{6-(-6)}$$
04
Calculate the slope of the original line
Computing the slope, we get:
$$m = \frac{-9}{12} = -\frac{3}{4}$$
So, the slope of the original line is \(-\frac{3}{4}\).
05
Find the slope of a perpendicular line
The slope of a line perpendicular to the original line will be the negative reciprocal of the original slope. So in this case:
$$m_{\perp} = \frac{4}{3}$$
Hence, the slope of the line perpendicular to the original line is \(\frac{4}{3}\).
06
Use point-slope form to write the equation of the perpendicular bisector
Using the point-slope form of a linear equation, which is:
$$y - y_m = m_{\perp}(x - x_m)$$
Substitute the values for \(x_m\), \(y_m\) and \(m_{\perp}\) from steps 2 and 5 and we get:
$$y - \left(-\frac{5}{2}\right) = \frac{4}{3}(x - 0)$$
07
Simplify the equation to get the final result
Simplifying the equation, we have:
$$y + \frac{5}{2} = \frac{4}{3}x$$
Multiplying both sides by \(6\) to eliminate fractions, we get:
$$6y + 15 = 8x$$
Rearrange the terms to obtain the equation of the perpendicular bisector:
$$8x - 6y = 15$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Midpoint
The midpoint of a line segment is a point that lies exactly halfway between two given endpoints. To find the midpoint, use the midpoint formula:
- For the x-coordinate: \( x_m = \frac{x_1+x_2}{2} \)
- For the y-coordinate: \( y_m = \frac{y_1+y_2}{2} \)
- \(x_m = \frac{-6 + 6}{2} = 0\)
- \(y_m = \frac{2 - 7}{2} = -\frac{5}{2}\)
Slope of a Line
The slope of a line measures its steepness or incline and is crucial for understanding the line's direction. Slope \( m \) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this exercise, using points \((-6, 2)\) and \((6, -7)\), substitute the values:
- \( m = \frac{-7 - 2}{6 - (-6)}\)
- \( m = \frac{-9}{12} = -\frac{3}{4} \)
Point-Slope Form
The point-slope form of a line's equation is very handy for quickly writing the equation of a line when you know a point on the line and its slope. The formula is:\[ y - y_1 = m(x - x_1) \]Where \( m \) is the slope, and \((x_1, y_1)\) is the known point. For the perpendicular bisector, our point is the midpoint \((0, -\frac{5}{2})\), and the perpendicular slope is \( \frac{4}{3} \). Plug these values in:
- \( y + \frac{5}{2} = \frac{4}{3}(x - 0) \)
Negative Reciprocal
A negative reciprocal is crucial when finding the slope of a line perpendicular to another. If you have the slope of a line, its perpendicular slope is the negative reciprocal of the original slope.
- Start with the original slope \(-\frac{3}{4}\).
- Flip the fraction to get \(\frac{4}{3}\).
- Change the sign to get \(\frac{4}{3}\).
