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Chapter 1: Problem 67

Exer. \(67-68\) : Find an equation for the perpendicular bisector of \(A B\). \(A(3,-1), B(-2,6)\)

Short Answer

Expert verified
The equation of the perpendicular bisector is \(y = \frac{5}{7}x + \frac{15}{14}\).

Step by step solution

01

Find the Midpoint of AB

To find the midpoint of the line segment \(AB\), use the midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1+y_2}{2} \right) \). Substituting \(A(3, -1)\) and \(B(-2, 6)\), we get the midpoint \( M = \left( \frac{3 + (-2)}{2}, \frac{-1 + 6}{2} \right) = \left( \frac{1}{2}, rac{5}{2} \right) \).
02

Determine the Slope of AB

The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the coordinates of \(A\) and \(B\), the slope \(m = \frac{6 - (-1)}{-2 - 3} = \frac{7}{-5} = -\frac{7}{5} \).
03

Calculate the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of \(AB\). Thus, it is \(-\left( -\frac{5}{7} \right) = \frac{5}{7}\).
04

Form the Equation of the Perpendicular Bisector

Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the midpoint, substitute the calculated slope \(\frac{5}{7}\) and point \(\left( \frac{1}{2}, rac{5}{2} \right)\). The equation becomes \(y - \frac{5}{2} = \frac{5}{7}\left(x - \frac{1}{2}\right)\). Simplify to obtain the equation: \[ y = \frac{5}{7}x + \frac{15}{14}\].

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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 vital tool in geometry, especially when dealing with line segments. It helps you find the exact middle point between two endpoints of a line segment. To apply the midpoint formula, you need the coordinates of both endpoints, say \( (x_1, y_1) \) and \( (x_2, y_2) \).
  • First, add the \( x \) coordinates of the endpoints and divide the sum by 2. This calculation gives the \( x \) coordinate of the midpoint.
  • Second, add the \( y \) coordinates and divide by 2 for the \( y \) coordinate of the midpoint.
For example, in the line segment \( AB \) with points \( (3, -1) \) and \( (-2, 6) \), the midpoint \( M \) is calculated as follows:
  • The \( x \) value is \( \frac{3 + (-2)}{2} = \frac{1}{2} \)
  • The \( y \) value is \( \frac{-1 + 6}{2} = \frac{5}{2} \)
Thus, the midpoint \( M \) of \( AB \) is \( \left( \frac{1}{2}, \frac{5}{2} \right) \). Finding the midpoint is essential in understanding not only line segments but also geometric shapes and their dimensions.
slope of a line
The slope of a line is a measure of its steepness and direction. It's calculated using the coordinates of two points on the line, let's call them \((x_1, y_1)\) and \((x_2, y_2)\). The formula for finding the slope (\( m \)) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here's a simple way to understand it:
  • The numerator \((y_2 - y_1)\) represents the change in vertical distance, or rise.
  • The denominator \((x_2 - x_1)\) represents the horizontal change, or run.
For the line \( AB \) with points \( (3, -1) \) and \( (-2, 6) \), the slope calculation follows:
  • \(y\) difference: \(6 - (-1) = 7\)
  • \(x\) difference: \(-2 - 3 = -5\)
  • Thus, \( m = \frac{7}{-5} = -\frac{7}{5} \)
This negative value indicates that as you move along the line from left to right, the line descends. The concept of slope is crucial in finding relationships between different lines, such as parallel and perpendicular lines.
point-slope form
The point-slope form is a handy tool for writing the equation of a line when you know its slope and a point on the line. The standard form is:\[y - y_1 = m(x - x_1)\]Here鈥檚 how it works:
  • \(m\) is the slope of the line.
  • \((x_1, y_1)\) is a known point on the line.
To use it, plug the known slope and point into the formula. For the perpendicular bisector of \( AB \), where the midpoint is \( \left( \frac{1}{2}, \frac{5}{2} \right) \) and the slope is \( \frac{5}{7} \), the equation is:
  • \(y - \frac{5}{2} = \frac{5}{7}(x - \frac{1}{2})\)
Solving this equation with algebraic manipulation, you'll get:
  • Multiply: \(5(x - \frac{1}{2})\) to achieve: \(y - \frac{5}{2} = \frac{5}{7}x - \frac{5}{14}\)
Understanding point-slope form allows you to quickly plunge into line-related problems. It simplifies the process when you don鈥檛 want to rearrange everything into slope-intercept form right away.
negative reciprocal
Understanding the negative reciprocal is key when dealing with perpendicular lines. If you know the slope of a line, the slope of a line perpendicular to it is the negative reciprocal of the original slope.Here's how it works:
  • Take the original slope, say \( m \).
  • Find its reciprocal by flipping the fraction (if it's \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \)).
  • Make it negative, so if the reciprocal is positive, it will become negative, and vice versa.
For line \( AB \) with slope \(-\frac{7}{5}\), the perpendicular bisector's slope is:
  • Reciprocal of \(-\frac{7}{5}\) is \(-\frac{5}{7}\)
  • Convert to positive: \(\frac{5}{7}\)
This concept is crucial to ensure lines meet perfectly at a right angle. No matter the original line's slope, you can always find the slope of a perpendicular line using this method.

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