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The problem of finding the perpendicular bisector of a line segment presents itself often in the study of analytic geometry. As with any problem of writing the equation of a line, you must determine the slope of the line and a point that the line passes through. A perpendicular bisector passes through the midpoint of the line segment and has a slope that is the negative reciprocal of the slope of the line segment. The problem can be solved as follows: Find the perpendicular bisector of the line segment between the points for the following. Write the equation in standard form. (a) \((-1,2)\) and \((3,0)\) Find the perpendicular bisector of the line segment between the points \((1,-2)\) and \((7,8)\). The midpoint of the line segment is \(\left(\frac{1+7}{2}, \frac{-2+8}{2}\right)\) \(=(4,3)\). The slope of the line segment is \(m=\frac{8-(-2)}{7-1}\) \(=\frac{10}{6}=\frac{5}{3}\). Hence the perpendicular bisector will pass through the point \((4,3)\) and have a slope of \(m=-\frac{3}{5}\). $$ \begin{aligned} y-3 &=-\frac{3}{5}(x-4) \\ 5(y-3) &=-3(x-4) \\ 5 y-15 &=-3 x+12 \\ 3 x+5 y &=27 \end{aligned} $$ Thus the equation of the perpendicular bisector of the line segment between the points \((1,-2)\) and \((7,8)\) is \(3 x+5 y=27\). (b) \((6,-10)\) and \((-4,2)\) (c) \((-7,-3)\) and \((5,9)\) (d) \((0,4)\) and \((12,-4)\)

Step by step solution

01

Calculate the Midpoint

To find the perpendicular bisector, first determine the midpoint of the line segment. For points \((6, -10)\) and \((-4, 2)\), the midpoint is calculated as follows:\[\left(\frac{6 + (-4)}{2}, \frac{-10 + 2}{2}\right) = \left(\frac{2}{2}, \frac{-8}{2}\right) = (1, -4)\].

02

Determine the Slope of the Original Line

The slope \(m\) of the line segment from \((6, -10)\) to \((-4, 2)\) is calculated by \[m = \frac{2 - (-10)}{-4 - 6} = \frac{2 + 10}{-10} = \frac{12}{-10} = -\frac{6}{5}\].

03

Calculate the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the original slope. So, for \(-\frac{6}{5}\), the perpendicular slope is \[m = \frac{5}{6}\].

04

Write the Equation of the Perpendicular Bisector

Using the point-slope form \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (1, -4)\) and \(m = \frac{5}{6}\), write the equation:\[y + 4 = \frac{5}{6}(x - 1)\].

05

Convert to Standard Form

Clear the fraction by multiplying every term by 6:\[6(y + 4) = 5(x - 1)\]which simplifies to:\[6y + 24 = 5x - 5\].Rearrange to the standard form:\[5x - 6y = 29\].

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Key Concepts

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

Analytic Geometry

Analytic geometry is the branch of mathematics that uses algebraic equations to describe geometric concepts. It helps us solve problems involving points, lines, circles, and other shapes. In the context of finding a perpendicular bisector, we use analytic geometry to calculate specific points and slopes that define the bisector line.
Here’s what you should know about analytic geometry when finding a perpendicular bisector:

• We work with coordinates to specify points on a plane.
• Equations help define lines through these points.
• Concepts such as slopes and midpoints are key to finding and describing lines.

Analytic geometry provides a concrete way to interact with abstract concepts by expressing them in numerical terms. This makes solving real-world geometric problems more straightforward.

Midpoint Formula

The midpoint formula is essential when working with a line segment in the coordinate plane. It allows us to find the exact center point of a segment defined by two endpoints. This is especially useful in determining the line path of a perpendicular bisector.To calculate the midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This will give you the "average" coordinate of the segment's endpoints.### Example If your points are \(6, -10\m\) and \(-4, 2\m\), apply the formula:\[\left( \frac{6 + (-4)}{2}, \frac{-10 + 2}{2} \right) = \left( \frac{2}{2}, \frac{-8}{2} \right) = (1, -4) \]This result, \(1, -4\m\), is the midpoint that the bisector will pass through.

Negative Reciprocal

A fundamental aspect of the concept of perpendicular lines is the idea of negative reciprocals. In analytic geometry, when two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other. This property helps determine the slope of the perpendicular bisector.### Understanding Negative Reciprocal

• If a line segment has a slope \(m\), the slope of a perpendicular line is \(-\frac{1}{m}\).
• This concept ensures that the two lines intersect at right angles.

### Example CalculationConsider a line segment with a slope of \(-\frac{6}{5}\). The perpendicular slope thus becomes \(\frac{5}{6}\). Understanding this relationship is crucial for determining the correct equation of the perpendicular bisector.

Standard Form Equation

The standard form equation of a line is expressed as \(Ax + By = C\), where \(A, B,\) and \(C\) are integers. Converting an equation to this form is often required when finalizing the expression of a line in analytic geometry.### How to ConvertWhen you have your equation in point-slope form, i.e., \(y - y_1 = m(x - x_1)\), follow these steps:

• Clear any fractions by multiplying each term by the denominator.
• Rearrange the terms to get all variables on one side of the equation.
• Ensure coefficients are integers.

### ExampleFor instance, convert \(y + 4 = \frac{5}{6}(x - 1)\) to standard form:1. Multiply through by 6 to eliminate fractions: \6(y + 4) = 5(x - 1)\.2. Distribute and rearrange: \5x - 6y = 29\.The result, \5x - 6y = 29\, is your standard form equation.