Problem 89 - 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 \(Q\). $$P(5,7), Q(2,11)$$

Step-by-Step Solution

Answer

Distance: 5 units, Midpoint: (3.5, 9)

Step 1: Understanding the Problem

We are given two points, \( P(5,7) \) and \( Q(2,11) \), and need to find the distance between them as well as the coordinates of the midpoint of the segment joining these two points.

Step 2: Calculating the Distance

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2)\) is given by the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the coordinates of \( P \) and \( Q \) into the formula, we get:\[d = \sqrt{(2 - 5)^2 + (11 - 7)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]Thus, the distance between \( P \) and \( Q \) is 5 units.

Step 3: Finding the Midpoint Coordinates

The midpoint \( M \) of a segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) has coordinates given by \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Using the coordinates of \( P \) and \( Q \), we find:\[\ M = \left( \frac{5+2}{2}, \frac{7+11}{2} \right) = \left( \frac{7}{2}, \frac{18}{2} \right) = \left( 3.5, 9 \right) \]Therefore, the coordinates of the midpoint \( M \) are \( (3.5, 9) \).

Key Concepts

Midpoint Formula

The midpoint formula is an essential tool in geometry when you're working with line segments on a coordinate plane. It helps you find the exact middle point of a line segment that connects any two points. The midpoint itself is simply the average of the starting and endpoint's coordinates.
When we say "midpoint," think of it as the point that divides the line segment into two equal parts. It answers the question of where the exact center of the segment is located. Here's an easy way to think about it:

• First, take the average of the x-coordinates of your points.
• Then, take the average of the y-coordinates.

The formula looks like this:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]So if you're given two points, say \( (5,7) \) and \( (2,11) \), you just plug these into the formula and it will tell you the midpoint's coordinates.
It’s pretty straightforward, and once you get the hang of it, finding midpoints becomes an easy task. It helps especially in applications like physics where you might need to determine a center point along a path.

Coordinates

Coordinates in mathematics are a way of pinpointing precise locations on a plane using two numbers, commonly known as ordered pairs. These numbers are usually represented as \((x, y)\), where:

• \(x\) indicates the horizontal distance from the origin.
• \(y\) indicates the vertical distance from the origin.

This system is known as the Cartesian coordinate system, named after the mathematician René Descartes. It allows us to map out almost any two-dimensional space, making it easier to analyze shapes, solve geometric problems, and even plot functions.For example, the point \( (5,7) \) tells us that you would move 5 units to the right (horizontally) and 7 units up (vertically) to reach your destination on the grid. These coordinates give us not just the location but also the ability to compute distances, slopes, and midpoints as in our exercise above.
The coordinate system is an essential part of geometry, physics, engineering, and many other fields that require spatial understanding and mathematical modeling.

Euclidean Geometry

Euclidean geometry is the study of plane and solid figures based on axioms and theorems employed by the ancient Greek mathematician Euclid. It’s the foundation for most of the work done in geometry today. At its heart, Euclidean geometry deals with relationships and properties of flat surfaces. This includes points, lines, planes, angles, and shapes like triangles, rectangles, and circles. All these elements and their interactions form the basis of Euclidean geometry.

• Points help us determine specific locations.
• Lines represent the shortest distance between two points.
• Planes can be thought of as endless flat surfaces extending in all directions.

In our exercise, finding the distance and midpoint between two points falls under this broad field of Euclidean geometry. It uses straightforward rules:

• The distance formula enables you to measure the length of a straight line between any two points.
• The midpoint formula helps determine the exact center of such a segment.

By understanding these principles, you can solve more complex geometric problems, learn more about the properties of various shapes, and even explore three-dimensional spaces.