Question

Prove that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) is $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$

Short answer

Question: Prove that the midpoint of the line segment joining two points P and Q with coordinates \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) in three-dimensional space has the coordinates \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)\). Answer: The midpoint of the line segment joining \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) is given by the coordinates $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$. This is proved by calculating the midpoint of each coordinate separately and then combining them.

Step-by-step solution

Understanding the problem

We are given two points P and Q with coordinates \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) in three-dimensional space. We are asked to prove that the midpoint of the line segment joining the two points has the coordinates \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)\).

Recall the definition of the midpoint

In any space, the midpoint of a line segment joining two points is a point that lies exactly halfway between the given points. In other words, it's a point that divides the line segment into two equal segments.

Find the midpoint for each coordinate

Now, we will find the midpoint of each coordinate separately (x-coordinate, y-coordinate, and z-coordinate) using the formula for the midpoint in each dimension, which is the average of the coordinates of the two points considered. Midpoint of x-coordinate: \(\frac{x_{1}+x_{2}}{2}\) Midpoint of y-coordinate: \(\frac{y_{1}+y_{2}}{2}\) Midpoint of z-coordinate: \(\frac{z_{1}+z_{2}}{2}\)

Combine the coordinates of the midpoint

Finally, to obtain the overall coordinates of the midpoint, we can combine the coordinates found in the previous step: $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$ Conclusion: We have shown that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}, z_{1}\right)\) and \(Q\left(x_{2}, y_{2}, z_{2}\right)\) is indeed given by the coordinates $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$.

Key concepts

Three-Dimensional Geometry

Three-dimensional geometry is a branch of geometry that deals with points, lines, planes, and shapes in three-dimensional space. Unlike two-dimensional planes with just length and width, three-dimensional space adds depth to these measurements. This additional dimension enables us to represent real-world objects more accurately, such as a sphere or a cube, since these objects not only have a length and width but also a height.

When working with three-dimensional geometry, you measure coordinates using three axes: the x-axis, y-axis, and z-axis. Each axis represents one of these dimensions. The calculation of the midpoint between two points is a common problem in this field because it allows us to understand symmetry and balance within three-dimensional objects. In essence, in three-dimensional geometry, we can consider an object in a more holistic way, accounting for all its possible edges, vertices, angles, and surfaces.

Mastering three-dimensional geometry helps with spatial reasoning, which is a crucial skill for fields such as engineering, architecture, and certain scientific disciplines.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This combination of algebra and geometry allows you to solve geometric problems by placing them in a numerical format. Using coordinates like \((x, y, z)\), we can precisely determine the location of points in three-dimensional space.

In coordinate geometry, the midpoint formula is a fundamental tool used to find the middle point of a line segment, offering a straightforward way to understand the symmetry between two points. To find the midpoint, you average the corresponding coordinates of the two points, as illustrated in the given exercise for the points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\).

• Midpoint of the x-coordinates is: \(\frac{x_1 + x_2}{2}\)
• Midpoint of the y-coordinates is: \(\frac{y_1 + y_2}{2}\)
• Midpoint of the z-coordinates is: \(\frac{z_1 + z_2}{2}\)

Each computation reflects combining algebra with geometric principles, demonstrating the power of coordinate geometry in solving spatial problems.

Proofs in Mathematics

Proofs are the backbone of mathematics, providing undeniable evidence that a statement or theorem is true. These logical arguments help to validate facts and establish trust in mathematical statements. In this specific context, proving the midpoint formula in three-dimensional space is an example of using a mathematical proof.
Proofs can vary from direct proofs, where you derive from known facts, to more complex structures like contradiction or induction. For the midpoint formula, we use a direct approach:

• Start with the definition of a midpoint, which is a point that divides a line segment into two equal parts.
• Apply this definition by averaging the coordinates of the given points.
• This simple calculation confirms that the resulting point lies exactly halfway between the two original points in each dimension.

Through mathematical proofs, we not only confirm the accuracy of formulas and concepts but also enhance our understanding of how they work. They deepen not only our comprehension but also contribute to the discipline's integrity.