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Chapter 2: Problem 24

Find a formula that expresses the fact that an arbitrary point \(P(x, y)\) is on the perpendicular bisector \(I\) of segment \(A B.\) $$A(-3,2), B(5,-4)$$

Short Answer

Expert verified
The equation of the perpendicular bisector is \(y = \frac{4}{3}x - \frac{7}{3}\).

Step by step solution

01

Calculate the Midpoint of Segment AB

To find the formula for the perpendicular bisector, we first need to determine the midpoint of segment \(AB\), which is the point equidistant from both \(A\) and \(B\). Use the midpoint formula \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). Substitute \(A(-3,2)\) and \(B(5,-4)\) into the formula: \[ M = \left( \frac{-3 + 5}{2}, \frac{2 + (-4)}{2} \right) = \left( 1, -1 \right) \] Thus, the midpoint of \(AB\) is \((1, -1)\).
02

Determine the Slope of Segment AB

Next, find the slope of segment \(AB\) using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute points \(A(-3,2)\) and \(B(5,-4)\) into the formula: \[ m_{AB} = \frac{-4 - 2}{5 - (-3)} = \frac{-6}{8} = -\frac{3}{4} \] The slope of \(AB\) is \(-\frac{3}{4}\).
03

Find the Slope of the Perpendicular Bisector

The perpendicular bisector of a line segment is perpendicular to the segment, so its slope is the negative reciprocal of the slope of \(AB\). Given \(m_{AB} = -\frac{3}{4}\), the slope of the perpendicular bisector \(m_I\) is \[ m_I = \frac{4}{3} \] Thus, the slope of the perpendicular bisector is \(\frac{4}{3}\).
04

Write the Equation of the Perpendicular Bisector

Using the point-slope form of a line \(y - y_1 = m(x - x_1)\), with \(m = \frac{4}{3}\) and using the midpoint \((1, -1)\) found in Step 1: \[ y + 1 = \frac{4}{3}(x - 1) \] Simplify to find the equation of the perpendicular bisector \(I\): \[ y + 1 = \frac{4}{3}x - \frac{4}{3} \] \[ y = \frac{4}{3}x - \frac{4}{3} - 1 \] \[ y = \frac{4}{3}x - \frac{7}{3} \] Thus, the equation of the perpendicular bisector is \(y = \frac{4}{3}x - \frac{7}{3}\).
05

Verify that Arbitrary Point \((x,y)\) Lies on the Perpendicular Bisector

A point \((x,y)\) lies on the perpendicular bisector if it satisfies the equation found in Step 4. For point \(P(x,y)\), verify it by substituting into the equation: \[ y = \frac{4}{3}x - \frac{7}{3} \] If the result is true, then point \(P\) is on the perpendicular bisector.

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Key Concepts

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

Midpoint Calculation
The midpoint is a crucial concept when dealing with line segments. It is the point exactly halfway between two endpoints of a segment. To calculate the midpoint, use the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This involves taking the average of the x-coordinates and the y-coordinates of the endpoints. For segment \( AB \) where \( A(-3,2) \) and \( B(5,-4) \), the calculations are straightforward:
  • Midpoint x-coordinate: \( \frac{-3 + 5}{2} = 1 \)
  • Midpoint y-coordinate: \( \frac{2 + (-4)}{2} = -1 \)
Thus, The midpoint \( M \) is \( (1, -1) \). This point becomes significant as it helps in deriving the slope and the equation of the perpendicular bisector.
Slope of a Line
The slope is a measure of how steep a line is. It is calculated as the change in y over the change in x. For a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula helps us find the slope of any line given two points.In our exercise, for segment \( AB \):
  • The change in y: \( -4 - 2 = -6 \)
  • The change in x: \( 5 - (-3) = 8 \)
  • Hence, slope \( m_{AB} = \frac{-6}{8} = -\frac{3}{4} \)
The slope tells us that for every 4 units you go right (positive x-direction), you go down 3 units (negative y-direction). This slope is vital for determining the slope of the perpendicular bisector later.
Point-Slope Form
The point-slope form is a way to express the equation of a line. It's particularly useful when you know a point on the line and the line's slope. The formula is:\[ y - y_1 = m(x - x_1) \]where \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope.To find the equation of the perpendicular bisector, we need the negative reciprocal of the slope of the original line. Since the slope of segment \( AB \) is \(-\frac{3}{4}\), the perpendicular bisector's slope \( m_I \) becomes \( \frac{4}{3} \). Using the midpoint \( (1, -1) \) as our point:
  • Substitute into the formula: \( y + 1 = \frac{4}{3}(x - 1) \)
This initial form is handy for writing the equation before simplifying it.
Equation of a Line
An equation of a line represents all points that lie on the line. For the perpendicular bisector, we follow a few algebraic steps to go from the point-slope form to the standard line equation, typically written as \( y = mx + b \).Starting from the point-slope form:
  • \( y + 1 = \frac{4}{3}(x - 1) \)
  • First, distribute the slope on the right: \( y + 1 = \frac{4}{3}x - \frac{4}{3} \)
  • Then, isolate \( y \) to get the line equation: \( y = \frac{4}{3}x - \frac{4}{3} - 1 \)
  • Finally simplify to get: \( y = \frac{4}{3}x - \frac{7}{3} \)
This equation represents the perpendicular bisector, detailing all \( (x, y) \) coordinates that form this line, confirming that it is the correct bisector of segment \( AB \).

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Most popular questions from this chapter

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