Problem 46 - Algebra and Trigonometry

Question

Plot the points \(P(-2,1)\) and \(Q(12,-1) .\) Which (if either) of the points \(A(5,-7)\) and \(B(6,7)\) lies on the perpendicular bisector of the segment \(P Q ?\)

Step-by-Step Solution

Answer

Point B(6, 7) lies on the perpendicular bisector of segment PQ.

Step 1: Calculate the Midpoint of Segment PQ

To find the midpoint of the segment connecting points \( P(-2, 1) \) and \( Q(12, -1) \), we use the midpoint formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).\]Substituting the coordinates of \( P \) and \( Q \), we get: \[\left( \frac{-2 + 12}{2}, \frac{1 + (-1)}{2} \right) = \left( \frac{10}{2}, \frac{0}{2} \right) = (5, 0).\]

Step 2: Calculate the Slope of Segment PQ

The slope of the line segment between \( P(-2, 1) \) and \( Q(12, -1) \) is calculated as: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{12 - (-2)} = \frac{-2}{14} = -\frac{1}{7}.\]

Step 3: Determine the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment. Thus, the slope of the perpendicular bisector is \[7\].

Step 4: Write the Equation of the Perpendicular Bisector

Using the midpoint \((5, 0)\) and the slope \(7\), we write the equation of the line using the point-slope form: \[y - y_1 = m(x - x_1),\]where \((x_1, y_1) = (5, 0)\) and \(m = 7\). So, \[y - 0 = 7(x - 5).\]Simplifying, we get:\[y = 7x - 35.\]

Step 5: Check if Point A Lies on the Perpendicular Bisector

Substitute point \(A(5, -7)\) into the equation \(y = 7x - 35\) to check:\[-7 = 7(5) - 35.\]Simplifying the right side gives:\[35 - 35 = 0,\]which is not equal to \(-7\). Thus, \(A\) does not lie on the line.

Step 6: Check if Point B Lies on the Perpendicular Bisector

Substitute point \(B(6, 7)\) into the equation \(y = 7x - 35\):\[7 = 7(6) - 35.\]Simplifying the right side gives:\[42 - 35 = 7,\]which matches the left side. Thus, point \(B\) does lie on the line.

Key Concepts

Midpoint Formula

The midpoint formula is a crucial tool for finding the center point of a line segment in coordinate geometry. To compute the midpoint of a segment defined by endpoints \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:

• Midpoint = \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

This formula simply averages the x-coordinates and the y-coordinates of the endpoints. In our example, the line segment connects points \(P(-2, 1)\) and \(Q(12, -1)\). By plugging these into our formula:

• \( \left( \frac{-2 + 12}{2}, \frac{1 + (-1)}{2} \right) = (5, 0) \)

This results in the midpoint of \(P Q\) being \( (5, 0) \). This point serves as the exact middle, which is essential for tasks like determining the perpendicular bisector.

Slope Calculation

The slope of a line indicates its steepness and direction. Slope is often denoted as "\( m \)" and calculated using the formula:

• Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this method, subtract the y-coordinates and x-coordinates of the points, then divide the differences. For the points \(P(-2, 1)\) and \(Q(12, -1)\), the slope is determined as follows:

• \( m = \frac{-1 - 1}{12 - (-2)} = \frac{-2}{14} = -\frac{1}{7} \)

This negative slope of \( -\frac{1}{7} \) indicates that the line goes downward from left to right. Understanding the slope is important for constructing perpendicular bisectors, as the slope of a perpendicular line is the negative reciprocal of the original.

Point-Slope Form

Point-slope form is another way to represent the equation of a line, particularly useful when you know a point on the line and its slope. The formula is:

• \( y - y_1 = m(x - x_1) \)

This approach simplifies the process of finding the linear equation by plugging in a specific point \( (x_1, y_1) \) and slope \( m \). Using the midpoint \( (5, 0) \) and slope \( 7 \) of the perpendicular bisector, the equation becomes:

• \( y - 0 = 7(x - 5) \)
• \( y = 7x - 35 \)

This tells us more about the line on which points are verified, helping identify positions in coordinate geometry such as which points lie on specific geometric constructs like perpendicular bisectors.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to use algebraic formulas to solve geometric problems on a coordinate plane. Key concepts include points, lines, and their relationships, assessed through calculations like the distance, midpoint, and slope. This discipline merges algebra's clear numerical properties with the visual understanding of geometry.

• Helps solve problems relating to shapes and measurements.
• Involves analyzing figures made up of points in the plane.

In the exercise example, determining the perpendicular bisector through calculations shows how the slope and point-slope form contribute to understanding geometric relationships and confirming if specific points \(A\) or \(B\) lie on such a bisector. It fuses abstract mathematical operations to portray very physical properties of lines and segments.