Q. 62 - Calculus

Question

Prove that the midpoint of a 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

The coordinates of midpoint of line segment \(L\) is \(\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2} \right)\).

Step 1. Given Information

Two points on a line \(L\) are  \(P\left(x_{1}, y_{1},z_{1}\right)\) and \(Q\left(x_{2}, y_{2},z_{2}\right)\) .

Step 2. Proof

Consider that the coordinates of the midpoint of the line segment L is \(\left(x,y,z\right)\).

Use the Distance Formula to calculate the distance of the point \(\left(x,y,z\right)\) 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 \(\left(x,y,z\right)\) from \(P\left(x_{1}, y_{1},z_{1}\right)\) is,

\(\sqrt{\left(x-x_{1}\right)^2+\left(y-y_{1}\right)^2 +\left(z-z_{1}\right)^2 }\)         .......\(\left(1\right)\)

The distance of the point \(\left(x,y,z\right)\) from \(Q\left(x_{2}, y_{2},z_{2}\right)\) is,

\(\sqrt{\left(x_{2}-x\right)^2+\left(y_{2}-y\right)^2 +\left(z_{2}-z\right)^2 }\)         .......\(\left(2\right)\)

Since a line segment \(L\) joins the points \(P\) and \(Q\), therefore, the point \(\left(x,y,z\right)\) is equidistant from the points \(P\) and \(Q\).

Thus, equate the distances of the point \(\left(x,y,z\right)\) 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 \(\left(1\right)\) and \(\left(2\right)\),

Along the x-axis,

\(x-x_{1}=x_{2}-x\)

\(2x=x_{2}+x_{1}\)

\(x=\frac{x_{1}+x_{2}}{2}\)

Along the y-axis,

\(y-y_{1}=y_{2}-y\)

\(2y=y_{2}+y_{1}\)

\(y=\frac{y_{1}+y_{2}}{2}\)

Along the z-axis,

\(z-z_{1}=z_{2}-z\)

\(2z=z_{2}+z_{1}\)

\(z=\frac{z_{1}+z_{2}}{2}\)

Thus, \(\left(x,y,z\right)=\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2} \right)\) 

Hence, the coordinates of the midpoint of the line segment \(L\) is \(\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2}\right)\).