Problem 5 - Calculus: One and Several Variables

Question

Find the midpoint of the line segment \(\overline{P_{0} P_{1}}\). $$P_{0}(2,4) . \quad P_{1}(6,8)$$

Step-by-Step Solution

Answer

The midpoint of the line segment \(\overline{P_0 P_1}\) is \(M(4,6)\).

Step 1: Identify the Coordinates

Identify the x and y coordinates for each point. For \(P_0(2,4)\), \(x_1 = 2\) and \(y_1 = 4\). For \(P_1(6,8)\), \(x_2 = 6\) and \(y_2 = 8\).

Step 2: Apply the Midpoint Formula

With the coordinates identified, the midpoint formula can be applied. That is, \(x_m = \frac{x_1 + x_2}{2}\) and \(y_m = \frac{y_1 + y_2}{2}\). Substituting the known values into the formula gives \(x_m = \frac{2 + 6}{2} = 4\) and \(y_m = \frac{4 + 8}{2} = 6\).

Step 3: Result

The calculated values represent the x and y coordinates of the midpoint. Therefore, the midpoint of the line segment \(\overline{P_0 P_1}\) is \(M(4,6)\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is like a map for math where points are plotted on a grid, also called a coordinate plane. Each point on this plane has its own address given by an

• x-coordinate: Shows how far left or right the point is from the origin (0,0).
• y-coordinate: Shows how far up or down the point is from the origin (0,0).

This mapping system helps students visualize and solve geometric problems using algebra. Coordinate Geometry is integral for finding distances and midpoints, and even understanding shapes like triangles and circles on a plane.
Whether you’re finding how two points relate to each other or discovering more about lines and angles, the coordinate plane is your go-to tool. It transforms math into a visual story, making concepts more engaging and easier to understand.

Line Segment

A line segment is a straight path connecting two endpoints. Think of it as a piece of a line that has a specific start and end point. In our midpoint exercise,

• \( \overline{P_{0} P_{1} } \): represents a line segment between point \( P_0 \) and point \( P_1 \).

It's important because a line segment is distinct from a line, which stretches infinitely in both directions. With line segments, we can actually make measurements, such as length, and identify specific points like the midpoint.
In real-world applications, line segments can represent anything from the edges of a shape to paths on a map. They’re a foundational element in geometry that you will often find in problems involving shapes and measurements.

Calculating Midpoints

Finding the midpoint of a line segment is like finding the average position between two points. This is useful when you want to know the exact middle of a path or connection. The midpoint formula is

• \( x_m = \frac{x_1 + x_2}{2} \): The horizontal midpoint between the two x-coordinates.
• \( y_m = \frac{y_1 + y_2}{2} \): The vertical midpoint between the two y-coordinates.

In the original exercise, by using the coordinates \( P_0(2,4) \) and \( P_1(6,8) \), you apply these formulas to find the midpoint, which is \( M(4,6) \).
This process involves adding the coordinates separately and dividing each sum by two. The outcome gives you the new x and y coordinates of the midpoint. Calculating midpoints is common in geometry for dividing objects into equal parts or finding the center of lines.