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Chapter 11: Problem 4

For the points \(A(4,7), B(28,11)\), and \(C(-3,-1)\), find the equation in point- slope form of the a. Perpendicular bisector of \(\overline{A B}\). (a) b. Median of \(\triangle A B C\) from point \(B\).

Short Answer

Expert verified
a. Perpendicular Bisector: \(y = -6(x-16) + 9\) b. Median from B: \(y = \frac{12}{31}x + \frac{-336}{31} + 11\)

Step by step solution

01

Find the Midpoint of Segment AB

To find the perpendicular bisector of segment \(\overline{AB}\), we first calculate the midpoint of \(AB\). The midpoint formula is: \[ \text{Midpoint} = \left( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} \right) \]Given points are \(A(4,7)\) and \(B(28,11)\). So the midpoint \(M_{AB}\) is:\[ M_{AB} = \left( \frac{4 + 28}{2}, \frac{7 + 11}{2} \right) = (16,9) \]
02

Find the Slope of Segment AB

The slope \(m\) of a line through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ m = \frac{y_2-y_1}{x_2-x_1} \]For \(A(4,7)\) and \(B(28,11)\), the slope of \(\overline{AB}\) is:\[ m_{AB} = \frac{11-7}{28-4} = \frac{4}{24} = \frac{1}{6} \]
03

Determine the Slope of the Perpendicular Bisector

The slope of two perpendicular lines are negative reciprocals. Thus, if \(m_{AB} = \frac{1}{6}\), the slope of the perpendicular bisector, \(m_{PB}\), is:\[ m_{PB} = -6 \]
04

Write the Equation of the Perpendicular Bisector

Using the point-slope form of the equation, \(y - y_1 = m(x - x_1)\), and the midpoint \(M_{AB} (16,9)\) as point \((x_1, y_1)\), and slope \(m = -6\):\[ y - 9 = -6(x - 16) \]This simplifies to:\[ y - 9 = -6x + 96 \]And further:\[ y = -6x + 105 \]So, the equation of the perpendicular bisector is:\[ y = -6(x-16) + 9 \]
05

Find the Coordinates of the Midpoint BC

To find the median from \(B\) to \(\triangle ABC\), calculate the midpoint of \(\overline{BC}\). Using \(B(28,11)\) and \(C(-3,-1)\), the midpoint \(M_{BC}\) is:\[ M_{BC} = \left( \frac{28 + (-3)}{2}, \frac{11 + (-1)}{2} \right) = \left( \frac{25}{2}, 5 \right) \]
06

Find the Slope of the Median from B

The slope of the median from \(B(28,11)\) to \(M_{BC}\) \(\left( \frac{25}{2}, 5 \right)\) is:\[ m = \frac{5 - 11}{ rac{25}{2} - 28} = \frac{-6}{ rac{-31}{2}} = \frac{-6}{- rac{31}{2}} = \frac{12}{31} \]
07

Write the Equation of the Median from B

Using the point-slope form, \(y - y_1 = m(x - x_1)\), with point \(B(28,11)\) and slope \(m = \frac{12}{31}\), we get:\[ y - 11 = \frac{12}{31}(x - 28) \]This simplifies to the median line equation:\[ y - 11 = \frac{12}{31}x - \frac{336}{31} \]Thus:\[ y = \frac{12}{31}x + \frac{-336}{31} + 11 \]

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Key Concepts

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

Perpendicular Bisectors
A perpendicular bisector is a line that divides a line segment into two equal parts at a right angle (90 degrees). This line is essential in coordinate geometry as it helps in finding equidistant points from the endpoints of a segment. To determine the equation of a perpendicular bisector, follow these steps:
  • First, find the midpoint of the segment. This midpoint will be a point through which the perpendicular bisector passes.
  • Find the slope of the original line segment. If the slope is known, the slope of the perpendicular bisector can be calculated as the negative reciprocal of this slope.
Once these calculations are made, use the point-slope form to write down the equation of the perpendicular bisector line.
Equations of Lines
Equations of lines form the backbone of geometry, defining how lines are positioned in a coordinate system. These equations can be assembled in different forms, such as point-slope and slope-intercept forms.
  • The point-slope form (\(y - y_1 = m(x - x_1)\)) is useful when the slope and a specific point on the line are known. It expresses a line's equation by connecting its slope to any known point.
  • Slope-intercept form (\(y = mx + b\)) is another popular format, highlighting how steep a line is and where it intersects the y-axis.
Using these forms, we can interchangeably express a line's equation to suit different problem-solving scenarios, like finding a perpendicular bisector or median in geometry problems.
Slope
The slope of a line showcases how steep the line is, representing the vertical change per unit of horizontal movement. It's a key component in determining the behavior of lines in the coordinate plane.
For a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it goes from left to right.
  • A slope of zero indicates a horizontal line, while an undefined slope (division by zero) represents a vertical line.
Understanding slope is crucial for analyzing line properties, such as creating perpendicular bisectors by finding negative reciprocals.
Midpoint Formula
The midpoint formula helps find the exact middle point of a segment that connects two points in a coordinate plane. This point is essential when looking to find bisectors or medians as it clearly defines the segment's center.
Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \((M_x, M_y)\) is calculated by:\[ (M_x, M_y) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]
  • This formula averages the x-coordinates and y-coordinates respectively, centralizing the segment.
  • It is key for constructing both perpendicular bisectors and medians, marking the point where calculations pivot.
Utilizing the midpoint formula helps inform further geometric constructions and symmetry studies.

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