Q. 62 - Calculus

Question

Prove that the midpoint of the line segment connecting the point $\left(x_1,y_1,z_1\right)$ to the point $\left(x_2,y_2,z_2\right)$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$.

Step-by-Step Solution

Answer

As a result, the coordinates of the line segment's midpoint $L$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$

Step 1: Introduction

Consider the following points:

$$\begin{gathered} P\left(x_1,y_1,z_1\right) \\ Q\left(x_2,y_2,z_2\right) \end{gathered}$$

Consider a line segment $L$ joining the points $P$ and $Q$.

The goal is to demonstrate that the line segment's midpoint is correct. $L$ is,

$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$

Step 2: Given information

Consider the coordinates of the line segment's midpoint. $L$ is $(x, y, z)$.

To calculate the distance between two points, use the Distance Formula.$(x, y, z)$ from the points $P\left(x_1,y_1,z_1\right)$ and $Q\left(x_2,y_2,z_2\right)$.

The distance of the point $(x, y, z)$ from the point $P\left(x_1,y_1,z_1\right)$ is,

$\sqrt{{\left(x-x_1\right)}^2+{\left(y-y_1\right)}^2+{\left(z-z_1\right)}^2}$

The distance of the point $(x, y, z)$ from the point $Q\left(x_2,y_2,z_2\right)$ is,

$\sqrt{{\left(x_2-x\right)}^2+{\left(y_2-y\right)}^2+{\left(z_2-z\right)}^2}$

$\sqrt{{\left(x_2-x\right)}^2+{\left(y_2-y\right)}^2+{\left(z_2-z\right)}^2}$

Step 3: Explanation

Since a line segment $L$ joins the points$P$ and $Q$, therefore, the point $(x, y, z)$ is equidistant

from the points $P$ and $Q$.

Thus, equate the distances of the point $(x, y, z)$ from the points $P\left(x_1,y_1,z_1\right)$ and

$Q\left(x_2,y_2,z_2\right)$ along the three axes.

From equation (1) and (2),

Along the $x$-axis,

$$\begin{gathered} x-x_1=x_2-x \\ 2x=x_2+x_1 \\ x=\frac{x_1+x_2}{2} \end{gathered}$$

Along the $y$-axis,

$\begin{array}{r}y-y_1=y_2-y\\ 2y=y_2+y_1\\ y=\frac{y_1+y_2}{2}\end{array}$

Along the$z$-axis,

$$\begin{gathered} z-z_1=z_2-z \\ 2z=z_2+z_1 \\ z=\frac{z_1+z_2}{2} \end{gathered}$$

Thus, $(x,y,z)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$.

As a result, the coordinates of the line segment's midpoint $L$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$