Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Find the coordinates of the midpoint of the line segment joining the two points. $$ (6,-9,1),(-2,-1,5) $$

Step-by-Step Solution

Answer

The coordinates of the midpoint of the line segment joining the points (6,-9,1) and (-2,-1,5) are (2, -5, 3).

Step 1: Understand the Midpoint Formula

In three-dimensional space, the midpoint M of two points A(x1, y1, z1) and B(x2, y2, z2) is given by the formula: M = \((x1+x2)/2, (y1+y2)/2, (z1+z2)/2\). So, to find the midpoint, you just have to add the corresponding coordinates of two points and divide by 2.

Step 2: Plug the Given Coordinates into the Midpoint Formula

The two given points are A(6, -9, 1) and B(-2, -1, 5). Plugging the coordinates into the formula gives M = \((6-2)/2, (-9-1)/2, (1+5)/2\).

Step 3: Simplify the Expressions

After performing the operations inside the parentheses in the last step, you get M = \((4)/2, (-10)/2, (6)/2\). This simplifies to M = \(2, -5, 3\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful branch of mathematics that allows us to analyze geometric properties and relationships using algebra and the Cartesian coordinate system. In this system, each point in space is defined by its coordinates, which are distances from the point to perpendicular reference lines called axes.

Specifically, in three-dimensional space (3D), a set of three numbers called coordinates \( (x, y, z) \) pinpoint the location of a point. The axes in 3D space are typically labeled as the x-axis (horizontal), y-axis (vertical on the plane), and z-axis (vertical into or out of the plane). The intersection of these three axes at the point \( (0, 0, 0) \) is known as the origin.

When dealing with geometry in 3D space, we can find distances, midpoints, and equations of lines and planes by applying algebraic methods to the coordinates of points. Understanding how to navigate this coordinate system is crucial for solving many types of problems, including the calculation of midpoints between two points in space.

3D Space Midpoint Calculation

The concept of finding a midpoint in a two-dimensional plane extends naturally into three dimensions. The midpoint of a line segment in 3D space is the point exactly halfway between its endpoints, and it's found by averaging the corresponding coordinates of these endpoints.

Let’s consider two points in 3D space, \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \). The midpoint formula in three dimensions is: \[ M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right) \].

This formula can be visualized as finding the average location between the two points along each axis. Once the midpoint is computed, it can be used for various applications, including computer graphics, engineering, and physics, where understanding the center point of a segment within 3D space is essential.

Algebraic Techniques

Algebraic techniques are the toolbox of methods and rules we use to manipulate algebraic expressions and solve equations. They include operations such as addition, subtraction, multiplication, division, and the use of the distributive property, among others.

In the context of finding the midpoint of two points in 3D space, the key algebraic technique involves applying the midpoint formula correctly, which requires basic arithmetic—adding or subtracting the corresponding coordinates of the two points and then dividing by two.

\textbf{Example:}To find the midpoint between \( A(6, -9, 1) \) and \( B(-2, -1, 5) \), we use the technique of averaging the coordinates. For the x-coordinates: \[ (6 - 2) / 2 = 2 \], y-coordinates: \[ (-9 - 1) / 2 = -5 \], and z-coordinates: \[ (1 + 5) / 2 = 3 \]. The final midpoint coordinates are \( M(2, -5, 3) \). Clearly understanding and applying these algebraic manipulations is critical to successfully solving problems in coordinate geometry.