Question

A line segment joins the points 8 − 5i and 2 + 9i on the complex plane. What are the length and the midpoint of the segment?

Answer 1

Given:

A line segment joins the points

$8-5i\text{ and 2+9i}$

To Find:

Length and midpoint of the segment.

Explanation:

Let the given points can be written as:

$\begin{gathered} a+bi=8-5i \\ s+ti=2+9i \end{gathered}$

To find the length:

Difference between the complex number is

$\begin{gathered} (8-5i)-(2+9i)=8-5i-2-9i \\ =6-14i \end{gathered}$$\begin{gathered} \text{Length=}\sqrt[]{6^2+(-14)^2} \\ =\sqrt[]{36+196} \\ =\sqrt[]{232} \\ =\sqrt[]{4\times58} \\ =2\sqrt[]{58}\text{ units} \end{gathered}$$\begin{gathered} \text{Midpoint of the line segment=}\frac{a+s}{2}+(\frac{b+t}{2})i \\ =\frac{8+2}{2}+(\frac{-5+9}{2})i \\ =\frac{10}{2}+(\frac{4}{2})i \\ =5+2i \end{gathered}$

Final answer:

$2\sqrt[]{58}\text{ ; 5+2i}$