Problem 52 - Algebra and Trigonometry

Question

Plot the points \(P(-2,1)\) and \(Q(12,-1)\) on a coordinate plane. 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 A(5, -7) lies on the perpendicular bisector of PQ.

Step 1: Find the Midpoint of PQ

The midpoint, M, of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated by the formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]. Substitute \( P(-2,1) \) and \( Q(12,-1) \) into the formula: \[ M = \left( \frac{-2 + 12}{2}, \frac{1 + (-1)}{2} \right) = (5, 0) \].

Step 2: Find the Slope of PQ

The slope \( m \) of a line that passes through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substitute \( P(-2,1) \) and \( Q(12,-1) \) into the formula: \[ m = \frac{-1 - 1}{12 - (-2)} = \frac{-2}{14} = -\frac{1}{7} \].

Step 3: Find the Slope of the Perpendicular Bisector

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus, if the slope of \( PQ \) is \(-\frac{1}{7}\), the slope of the perpendicular bisector is \( 7 \).

Step 4: Write the Equation of the Perpendicular Bisector

Using the point-slope form of a linear equation \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the midpoint \(M(5,0)\) and the slope \(m = 7\), the equation is: \[ y - 0 = 7(x - 5) \], which simplifies to \[ y = 7x - 35 \].

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

Substitute point \( A(5, -7) \) into the equation \( y = 7x - 35 \): \(-7 = 7(5) - 35\), which simplifies to \(-7 = 35 - 35 \). This equation holds true, so point \( A \) lies on the perpendicular bisector.

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

Substitute point \( B(6, 7) \) into the equation \( y = 7x - 35 \): \( 7 = 7(6) - 35 \), which simplifies to \( 7 = 42 - 35 \). This does not hold true as \( 7 eq 7 \), so point \( B \) does not lie on the perpendicular bisector.

Key Concepts

Midpoint Formula

The midpoint formula is a crucial tool in coordinate geometry. It allows you to find the center point of a line segment that connects two points in a coordinate plane. To calculate the midpoint of a line segment, you simply take the average of the x-coordinates and the average of the y-coordinates of the endpoints.
For instance, if you have the points \( (-2, 1) \) and \( (12, -1) \), the midpoint \( M \) is given by the formula:

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

This midpoint is particularly significant when you're working on problems involving symmetry and bisectors. It provides a balanced and central position along the segment, serving as a reference point for other calculations.

Slope of a Line

The slope of a line is a measure of its steepness or angle. It describes the rate at which y-values change relative to x-values along the line. The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated with the formula:

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

For the segment connecting points \((-2, 1)\) and \( (12, -1)\), the slope is:

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

The slope indicates whether a line is inclined upwards, downwards, or is horizontal. A negative slope means the line descends from left to right. This concept forms the foundation for many advanced topics, such as determining parallel and perpendicular lines.

Perpendicular Bisector

A perpendicular bisector is a line that cuts another line segment into two equal parts at a right angle. In coordinate geometry, finding the perpendicular bisector involves using the midpoint as a reference and calculating the slope that is the negative reciprocal of the original line's slope.
For a line segment with a slope of \(-\frac{1}{7}\), the slope of its perpendicular bisector becomes \(7\).
This relationship is fundamental in constructing bisectors for various geometric applications, such as triangle circumcenters or simulating symmetry. By incorporating the midpoint \((5, 0)\) into the point-slope form, we get the equation of the perpendicular bisector:

• \( y - 0 = 7(x - 5) \).

This simplifies to:

• \( y = 7x - 35 \).

This equation defines the set of all points that are equidistant from the segment's endpoints and are aligned perpendicularly.

Point-Slope Form

The point-slope form is a convenient way of writing the equation of a line when you know a point on the line and its slope. The general form is expressed as:

• \( y - y_1 = m(x - x_1) \).

Here, \((x_1, y_1)\) is a known point on the line, and \( m \) is the slope.
For example, using the midpoint \((5, 0)\) and a slope \(7\), the equation of the perpendicular bisector obtained is

• \( y - 0 = 7(x - 5) \).

This simplifies to:

• \( y = 7x - 35 \).

Understanding the point-slope form is essential for writing the equation of a line directly and efficiently. It is particularly helpful for geometrical constructions such as drawing bisectors, parallels, or tangents.