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Chapter 2: Problem 21

Prove that \(C\) is on the perpendicular bisector of segment \(A B\) $$A(-4,-3), \quad B(6,1), \quad C(5,-11)$$

Short Answer

Expert verified
Point C is on the perpendicular bisector of segment AB.

Step by step solution

01

Find the Midpoint of Segment AB

The midpoint of a segment with endpoints \((x_1,y_1), (x_2, y_2)\) is given by the formula: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \.\] Plugging in the coordinates of points \(A\) and \(B\), we have \[ \left( \frac{-4 + 6}{2}, \frac{-3 + 1}{2} \right) = (1, -1).\] The midpoint of \(AB\) is thus \((1, -1)\).
02

Calculate the Slope of Segment AB

The slope of a line through two points \((x_1, y_1), (x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \.\] For segment \(AB\), calculate \[ m = \frac{1 - (-3)}{6 - (-4)} = \frac{1 + 3}{6 + 4} = \frac{4}{10} = \frac{2}{5}.\] The slope of \(AB\) is \(\frac{2}{5}\).
03

Determine the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of \(AB\). Since the slope of \(AB\) is \(\frac{2}{5}\), the slope of the perpendicular bisector is \(-\frac{5}{2}\).
04

Verify that Point C Lies on the Perpendicular Bisector

To check if \(C\) lies on the line with slope \(-\frac{5}{2}\) and passing through the midpoint \((1, -1)\), use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1), \] where \(m = -\frac{5}{2}, (x_1, y_1) = (1, -1)\). This gives \[ y + 1 = -\frac{5}{2}(x - 1). \] Simplifying, \[ y + 1 = -\frac{5}{2}x + \frac{5}{2}. \] Therefore, \[ y = -\frac{5}{2}x + \frac{5}{2} - 1, \] \[ y = -\frac{5}{2}x + \frac{3}{2}. \] Substitute \(x = 5\) and \(y = -11\) to check point \(C\): \[ -11 = -\frac{5}{2}(5) + \frac{3}{2}. \] This simplifies to \[ -11 = -\frac{25}{2} + \frac{3}{2} = -\frac{22}{2} = -11.\] Thus, \(C(5, -11)\) satisfies the line equation.

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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 simple tool in coordinate geometry to find the center point of a line segment. Imagine you have a segment with two given endpoints. The midpoint is located exactly halfway, right between these two endpoints.

To find this midpoint, use the formula:
  • Take the average of the x-coordinates: \(x_{mid} = rac{x_1 + x_2}{2}\)
  • Take the average of the y-coordinates: \(y_{mid} = rac{y_1 + y_2}{2}\)
Let's see how it works with our exercise. If point A is \( (x_1, y_1) = (-4, -3) \) and point B is \( (x_2, y_2) = (6, 1)\), calculating the midpoint gives:
  • Midpoint x-coordinate: \( rac{-4 + 6}{2} = 1 \)
  • Midpoint y-coordinate: \( rac{-3 + 1}{2} = -1 \)
So, the midpoint of segment AB is \( (1, -1) \). It provides the balance or center of the segment.
Slope Calculation
Understanding the slope of a line is fundamental in coordinate geometry. It tells you how steep the line is. By calculating the slope, you discover how much the y-coordinate changes with the x-coordinate.

The formula to find the slope \( m \) between two points is:
  • Vertical change: \( ext{rise} = y_2 - y_1 \)
  • Horizontal change: \( ext{run} = x_2 - x_1 \)
  • Slope: \( m = rac{ ext{rise}}{ ext{run}} \)
For segment AB in our exercise, with points A(-4,-3) and B(6,1), the slope is calculated as:
  • Rise: \( 1 - (-3) = 4 \)
  • Run: \( 6 - (-4) = 10 \)
  • Slope: \( m = rac{4}{10} = rac{2}{5} \)
This slope \( rac{2}{5} \) means that for every 5 units you move horizontally, you move 2 units vertically.
Perpendicular Bisector
A perpendicular bisector is a line that divides another line segment into two equal parts at a right angle. Why is it so special? Because if another point is equidistant from the endpoints of this segment, it will lie on this bisector.

To find the equation of a perpendicular bisector:
  • Determine the midpoint of the segment using the midpoint formula. We've already done this, finding it to be \( (1, -1) \).
  • Find the slope of the original segment. For our segment AB, it鈥檚 \( rac{2}{5} \).
  • The perpendicular bisector has the negative reciprocal slope. So, switch the numerator with the denominator and change the sign: \( - rac{5}{2} \).
Thus, the perpendicular bisector of AB will intersect AB at \( (1, -1) \) and have a slope of \( - rac{5}{2} \). Any point on this line will be at an equal distance from points A and B.
Equation of a Line
The equation of a line, especially in point-slope form, is crucial for showing where a line goes on a graph given one point and its slope.

To form a line equation, we use:
  • Point-Slope Formula: \( y - y_1 = m(x - x_1) \)
  • Here, \( (x_1, y_1) \) is a known point, and \( m \) is the slope.
From our exercise, the midpoint is \( (1, -1) \) and the slope for the perpendicular bisector is \( -\frac{5}{2} \). Plugging these into the formula:
  • Starting Point-Slope: \( y + 1 = -\frac{5}{2}(x - 1) \)
  • Solving gives: \( y = -\frac{5}{2}x + \frac{3}{2} \)
We can use this line equation to verify if point C(5,-11) lies on it. Substituting x and y:
  • If both sides equal, then C is on the line, confirming its location on the bisector.
And as calculated, point C does indeed lie on this line.

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