Problem 108 - A Graphical Approach to College Algebra

Question

Solve each problem. Prove that the midpoint \(M\) of the line segment joining endpoints \(P\left(x_{1}, y_{1}\right)\) and $Q\left(x_{2}, y_{2}\right) has coordinates $$ \left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) $$ by showing that the distance between P and M is equal to the distance between M and Q and that the sum of these distances is equal to the distance between P and Q.

Step-by-Step Solution

Answer

The distances PM = MQ, and PM + MQ = PQ, proving the midpoint formula is correct.

Step 1: Identify Midpoint Formula

The midpoint, \( M \), of a line segment with endpoints \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is calculated using the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] We are tasked with proving that using this formula will result in a point \( M \) such that the distance \( PM = MQ \) and the sum \( PM + MQ = PQ \).

Step 2: Calculate Distances

First, calculate the distance \( PM \) using the distance formula: \[ PM = \sqrt{\left( \frac{x_1 + x_2}{2} - x_1 \right)^2 + \left( \frac{y_1 + y_2}{2} - y_1 \right)^2} \]This simplifies to: \[ PM = \sqrt{\left( \frac{x_2 - x_1}{2} \right)^2 + \left( \frac{y_2 - y_1}{2} \right)^2} \]Using the same procedure, calculate \( MQ \): \[ MQ = \sqrt{\left( x_2 - \frac{x_1 + x_2}{2} \right)^2 + \left( y_2 - \frac{y_1 + y_2}{2} \right)^2} \]This simplifies to: \[ MQ = \sqrt{\left( \frac{x_2 - x_1}{2} \right)^2 + \left( \frac{y_2 - y_1}{2} \right)^2} \]

Step 3: Prove PM Equals MQ

Notice that both \( PM \) and \( MQ \) are expressed by the same formula: \[ \sqrt{\left( \frac{x_2 - x_1}{2} \right)^2 + \left( \frac{y_2 - y_1}{2} \right)^2} \]Thus, we have shown \( PM = MQ \) as required.

Step 4: Calculate Distance PQ

Now, calculate the whole length \( PQ \), the distance from \( P \) to \( Q \): \[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 5: Prove Distance Sum

Given our expressions for \( PM \) and \( MQ \), we can note:\[ PM + MQ = 2 \times \sqrt{\left( \frac{x_2 - x_1}{2} \right)^2 + \left( \frac{y_2 - y_1}{2} \right)^2} \]By recognizing that this simplifies to:\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = PQ \] Therefore:\[ PM + MQ = PQ \] This confirms the midpoint result, \( PM + MQ = PQ \), proving correctness.

Key Concepts

Distance Formula

The concept of the Distance Formula is integral to understanding how to measure the length between two points in a plane. In coordinate geometry, points are defined by coordinates, for instance, point \(P(x_1, y_1)\) and point \(Q(x_2, y_2)\). The distance formula is formulated from the Pythagorean theorem and calculates the straight line distance – a measure of the 'as-the-crow-flies' distance – between these two points.
\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula allows for consistent calculations across all types of coordinate planes, whether they are Cartesian or any variant with axes. The formula's square root function interprets the algebraic difference in both \(x\) and \(y\) dimensions to produce a single scalar value, which is the length of the line segment between the two points. When using the Distance Formula to verify midpoint relationships like in our original exercise, it acts as a mathematical validator to ensure that segment breakups respect geometric symmetry, as every line segment from a midpoint to its respective endpoint should naturally be the same length.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints. It differs from a line in geometry as it has a definite start and end, which means it doesn't extend infinitely in either direction.
In coordinate geometry, line segments can be defined by their endpoints, often expressed as \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). Because endpoints are fixed, line segments have lengths that can be calculated with the Distance Formula.
This makes the concept of line segments vital when exploring relationships like midpoints, particularly in calculating if segment halves are equivalent. The importance of line segments in geometry is highlighted by their presence in various functions, slopes, and geometric figures, stored with definite measures and roles in assisting with deriving perimeters, areas, and in this case, small heroes in proof by calculation and demonstration.
In our exercise, validating that the midpoint divides the line segment into two equal parts is crucial, showing not just the measure but symmetry of these line segments.

Coordinate Geometry

Coordinate Geometry, often referred to as analytical geometry, is the study of geometry using the coordinate system. It merges algebra and Euclidean geometry to describe geometric shapes numerically and locate points in a plane.
In a typical Cartesian coordinate system, points are described by pairs \( (x, y) \), allowing for simple calculations of distances and midpoints. Coordinate geometry provides distinct advantages in geometry problems by converting geometric conditions into algebraic equations and inequalities, which are more straightforward to solve or prove.
Using coordinate geometry, as in our example, provides a structured method to derive attributes of geometric figures like line segments – including calculating midpoints or proving properties such as sum of segments or concretely exhibiting symmetry.
Here, the task was to prove that two halves created by the midpoint are equal using coordinate geometry's tools: establishing midpoint's coordinates and employing the Distance Formula to demonstrate dual segment equality. Thus, this branch of geometry serves as a powerful tool for students to bridge conceptual and computational understanding in their mathematical tasks.