A segment in the complex plane has a midpoint at 3 – 2i. If one endpoint of the segment is at 7 + i, what is the other endpoint?

2 answers:

7 0

Well, we know the midpoint is 3 - 2i.. well, skipping for a second the 'i', is just (3, - 2) then

one endpoint is at 7 + i, or 7 + 1i or , ( 7, 1)

let's say the other endpoint is at x,y

$\bf \textit{middle point of 2 points }\\ \quad \\
\begin{array}{lllll}
&x_1&y_1&x_2&y_2\\
% (a,b)
&({{ 7}}\quad ,&{{ 1}})\quad 
% (c,d)
&({{ x}}\quad ,&{{ y}})
\end{array}\qquad
% coordinates of midpoint 
\left(\cfrac{{{ x_2}} + {{ x_1}}}{2}\quad ,\quad \cfrac{{{ y_2}} + {{ y_1}}}{2} \right)
\\\\\\
\left(\cfrac{{{ x}} + {{ 7}}}{2}\quad ,\quad \cfrac{{{ y}} + {{ 1}}}{2} \right)=(3,-2)\implies 
\begin{cases}
\frac{x+7}{2}=3\\
x+7=6\\
x=6-7\\
------\\
\frac{y+1}{2}=-2\\
y+1=-4\\
y=-4-1
\end{cases}$

so, x = -1 and y = -5, that gives us ( -1, -5) or -1 - 5i on the imaginary grid