Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Find (a) the distance between \(P\) and \(Q\) and (b) the coordinates of the midpoint \(M\) of the segment joining \(P\) and \(Q\). $$P(-2,5), Q(4,-3)$$

Step-by-Step Solution

Answer

Distance is 10, and the midpoint is (1, 1).

Step 1: Identify Coordinates

The coordinates of point \(P\) are \((-2, 5)\) and the coordinates of point \(Q\) are \((4, -3)\). These points will be used to calculate both the distance and the midpoint of the segment joining them.

Step 2: Calculate Distance Using Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the coordinates of \(P\) and \(Q\) into the formula:\[ d = \sqrt{(4 - (-2))^2 + (-3 - 5)^2} \]Simplify:\[ d = \sqrt{(4 + 2)^2 + (-8)^2} \]\[ d = \sqrt{6^2 + (-8)^2} \]\[ d = \sqrt{36 + 64}\]\[ d = \sqrt{100} \]\[ d = 10 \]

Step 3: Find Midpoint Using Midpoint Formula

The midpoint \((x_m, y_m)\) of a segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the midpoint formula: \[ (x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Substitute the coordinates of \(P\) and \(Q\):\[ (x_m, y_m) = \left( \frac{-2 + 4}{2}, \frac{5 - 3}{2} \right) \]Simplify:\[ (x_m, y_m) = \left( \frac{2}{2}, \frac{2}{2} \right) \]\[ (x_m, y_m) = (1, 1) \]

Key Concepts

Midpoint Formula

The midpoint formula is a straightforward yet powerful tool used in coordinate geometry to find the center point between two coordinates. When you have two points, \(x_1, y_1\) and \(x_2, y_2\), on a coordinate plane, the midpoint formula helps you calculate the average location between these two points. It's like finding the exact halfway mark on a map between two cities. The formula is written as: \[ (x_m, y_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula simply averages the x-coordinates and the y-coordinates separately to provide a new pair of coordinates that represents the middle point. This point, also called the midpoint, can be visualized as the balancing point if the line segment between the two points were a seesaw. Consider it like sharing a pizza between two friends equally, where the midpoint ensures both have exactly half. It can be particularly handy in graphic design, mathematics, navigation, and many fields where precise positioning is vital.

Coordinate Geometry

Coordinate geometry, also famously termed 'analytic geometry,' bridges algebra and geometry by using coordinates to describe geometric shapes. This branch of mathematics enables us to use algebraic methods to solve geometric problems by converting them into equations. Here's why it's important:

• **Precision**: It allows for exact placement and measurement of points, lines, and shapes using numerical coordinates.
• **Relationship**: Demonstrates how algebraic equations can define geometric shapes and vice versa.
• **Application**: It has practical applications in various tasks, such as plotting curves and lines on graphs, defining the layout of architectural plans, and more.

In coordinate geometry, each point on a plane is defined by an x and y coordinate, making it possible to perform numerical operations on them. This means you can apply formulas such as the distance formula and the midpoint formula to calculate lengths, distances, areas, and more. It's like turning the real world scenarios, like roads and buildings, into equations on paper or a computer screen, thus allowing for extensive analysis and design.

Precalculus

Precalculus serves as the groundwork for calculus, providing students with the essential concepts needed before diving into the study of change and motion that calculus offers. It's like preparing the soil before planting a tree; you need this firm foundation for the upcoming complex concepts. Essential components of precalculus include:

• **Functions**: Understanding how to evaluate and interpret different types of functions, such as linear, polynomial, and logarithmic.


• **Algebraic Skills**: Improving skills in solving equations, factoring, and working with exponents and radicals.


• **Trigonometry**: Introducing the basics of sine, cosine, tangent, and their applications.


• **Graphs and Models**: Learning how to plot functions and interpret graphs critically.

These aspects are not just academic pursuits; they are used in solving real-world problems related to physics, engineering, economics, and more. By mastering precalculus, students become equipped to handle the rigorous challenges of calculus, ultimately enhancing their analytical and problem-solving skills. Precalculus uses tools like the distance and midpoint formulas to introduce students to deeper mathematical territories.