Question

Point K is the midpoint of QZ. Point K is located at (-1,-11), and point Z is located at (7,-3). Where is point Q located?

Answer 1

`\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ Q(\stackrel{x_1}{x}~,~\stackrel{y_1}{y})\qquad Z(\stackrel{x_2}{7}~,~\stackrel{y_2}{-3}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{7+x}{2}~~,~~\cfrac{-3+y}{2} \right)~~=~~\stackrel{\stackrel{midpoint}{K}}{(-1,-11)}\implies \begin{cases} \cfrac{7+x}{2}=-1\\[1em] 7+x=-2\\ \boxed{x=-9}\\ \cline{1-1} \cfrac{-3+y}{2}=-11\\[1em] -3+y=-22\\ \boxed{y=-19} \end{cases}`

Answer 2

`(-9,-19)`

Explanation:

Givens

`K(-1,-11)\\Z(7,-3)\\Q(x_(1),y_(1))`

Now, the definition of midpoint is

`K((x_(1)+x_(2) )/(2) ,(y_(1)+y_(2))/(2) )`

But, we know that
`K(-1,-11)`, so we replace each vale, where
`x_(2) =7` and
`y_(1)=-3`

`-1=(x_(1)+7 )/(2) \\-2=x_(1)+7\\-2-7=x_(1)\\x_(1)=-9\\`

`(y_(1)+y_(2))/(2)=-11\\y_(1)-3=-22\\y_(1)=-22+3\\y_(1)=-19`

Therefore, Q is located at
`(-9,-19)`