Problem 94 - 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 \(Q\). $$P(8.9,1.6), Q(3.9,13.6)$$

Step-by-Step Solution

Answer

(a) Distance = 13 units; (b) Midpoint = (6.4, 7.6).

Step 1: Understanding the Distance Formula

The distance between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) in the Cartesian plane is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substitute the coordinates of points \(P\) and \(Q\) into the formula.

Step 2: Calculate the Difference in Coordinates

Substitute the coordinates of \(P(8.9, 1.6)\) and \(Q(3.9, 13.6)\) into the distance formula: \[ x_2 - x_1 = 3.9 - 8.9 = -5 \]\[ y_2 - y_1 = 13.6 - 1.6 = 12 \]

Step 3: Plug Values into Distance Formula

Now substitute the differences into the distance formula:\[ d = \sqrt{(-5)^2 + (12)^2} \] \[ d = \sqrt{25 + 144} \] \[ d = \sqrt{169} \]

Step 4: Solve for Distance

Simplify the radical to find the distance:\[ d = 13 \] Thus, the distance between points \( P \) and \( Q \) is 13 units.

Step 5: Understanding the Midpoint Formula

The coordinates of the midpoint \( M \) of a segment joining two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) are given by:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substitute the coordinates of points \( P \) and \( Q \) into the formula.

Step 6: Calculate the Coordinates of the Midpoint

Substitute the coordinates of \(P(8.9, 1.6)\) and \(Q(3.9, 13.6)\) into the midpoint formula:\[ M_x = \frac{8.9 + 3.9}{2} = 6.4 \]\[ M_y = \frac{1.6 + 13.6}{2} = 7.6 \] Thus, the coordinates of the midpoint \( M \) are \( (6.4, 7.6) \).

Key Concepts

Midpoint Formula

When you're asked to find the midpoint of a line segment between two points in the Cartesian plane, the midpoint formula is your go-to tool. It's like finding the average location between two points. Imagine you're halfway through a journey, and you want to know exactly where you are.

The midpoint formula is expressed as:

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

Simply take the average of the x-coordinates and the average of the y-coordinates.
This will give you the coordinates of the midpoint. For instance, with points \( P(8.9, 1.6) \) and \( Q(3.9, 13.6) \), the midpoint \( M \) is \( (6.4, 7.6) \).

Breaking it down:

• For the x-coordinate: Add 8.9 and 3.9, then divide by 2 to get 6.4.
• For the y-coordinate: Add 1.6 and 13.6, then divide by 2 to get 7.6.

The midpoint signifies a perfect balance between two points, making it an invaluable concept in coordinate geometry.

Cartesian Plane

The Cartesian plane is a two-dimensional plane where each point is uniquely defined by a pair of numerical coordinates. These coordinates are obtained by the intersection of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

Think of the Cartesian plane as a map where every location can be pinpointed using two numbers:

• The x-coordinate tells you how far to move horizontally from the origin (0,0).
• The y-coordinate tells you how far to move vertically.

The Cartesian plane is fundamental for graphing and analyzing functions, geometry, and solving equations.

In the context of our original exercise, points \( P(8.9, 1.6) \) and \( Q(3.9, 13.6) \) lie on this plane. The x and y coordinates help to precisely define their positions.

This plane makes it not only possible to visually represent relationships between numbers but also to understand spatial relationships and distances, like calculating how far two points are from each other using the distance formula.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry using the Cartesian coordinate system. This branch of mathematics allows you to describe geometric figures algebraically and depict equations graphically.

How does this work in practice? Let's break it down:

• The position of points, lines, and curves can be described using numbers (coordinates).
• Geometric problems can be solved using algebraic techniques.

For example, in our exercise, we used coordinate geometry principles to find the distance and midpoint between points \( P \) and \( Q \).
You identify each point with its respective coordinates and go on to calculate properties such as distance using the distance formula and midpoints using the midpoint formula.

Coordinate geometry is essential because it combines the visual aspect of geometry with the computational power of algebra. It allows solving a multitude of geometric problems and is widely applied in fields like physics, engineering, and computer graphics.