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.

Step-by-step 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*58} \\ =2\sqrt[]{58}\text{ units} \end{gathered}`
`\begin{gathered} \text{Midpoint of the line segment=}(a+s)/(2)+((b+t)/(2))i \\ =(8+2)/(2)+((-5+9)/(2))i \\ =(10)/(2)+((4)/(2))i \\ =5+2i \end{gathered}`

Final answer:

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