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Magazine Chapter 3: Problem 92
Find an equation of the perpendicular bisector of the line segment whose endpoints are given. \((-6,-3) ;(-8,-1)\)
Short Answer
Expert verified
The equation of the perpendicular bisector is \(y = x + 5\).
Step by step solution
01
Find the Midpoint
The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). For the points \((-6, -3)\) and \((-8, -1)\), calculate \(\left( \frac{-6 + (-8)}{2}, \frac{-3 + (-1)}{2} \right) = \left( \frac{-14}{2}, \frac{-4}{2} \right) = (-7, -2)\). Therefore, the midpoint is \((-7, -2)\).
02
Calculate the Slope of the Original Line
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For our points, compute \(m = \frac{-1 - (-3)}{-8 - (-6)} = \frac{-1 + 3}{-8 + 6} = \frac{2}{-2} = -1\). The slope of the original line is \(-1\).
03
Determine the Perpendicular Slope
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. For a slope of \(-1\), the perpendicular slope is \(1\) (since \(-1\) becomes \(\frac{1}{1}\), and its negative reciprocal is \(1\)). So, the perpendicular slope is \(1\).
04
Write the Equation of the Perpendicular Bisector
Using the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the midpoint \((-7, -2)\) and \(m\) is the perpendicular slope \(1\), substitute to get \(y - (-2) = 1(x - (-7))\). Simplify to \(y + 2 = x + 7\). Rearrange to get the equation of the perpendicular bisector: \(y = x + 5\).
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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 mathematical tool used to find the exact middle point between two endpoints of a line segment. This point is called the midpoint. To calculate it, we use the formula \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
- The term \((x_1, y_1)\) represents the coordinates of the first point of the line segment.
- The term \((x_2, y_2)\) represents the coordinates of the second point of the line segment.
Slope of a Line
The slope of a line describes how steep the line is and in which direction it tilts. It is a measure of the line鈥檚 inclination from the x-axis and is calculated using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
- \(y_2 - y_1\) is the difference in the y-coordinates of two points.
- \(x_2 - x_1\) is the difference in the x-coordinates of these two points.
Negative Reciprocal
The negative reciprocal of a number is found by inverting the number and then changing its sign. When two lines are perpendicular, their slopes are negative reciprocals of each other.To find the negative reciprocal of a slope:
- First, identify the original slope \(m\).
- Then, take the reciprocal, \(\frac{1}{m}\).
- Finally, change the sign: if \(m\) is positive, make it negative and vice versa.
Point-Slope Form
The point-slope form is a linear equation tool used to pinpoint a line when you know one point on the line and its slope. The equation is given by\[y - y_1 = m(x - x_1)\]
- \((x_1, y_1)\) is a known point on the line, typically the midpoint in problems involving perpendicular bisectors.
- \(m\) represents the slope of the line.
