If the midpoint of the line joining (1 - k, -4) and (2, k + 1) is (-k, k), find the value of k.
Answer Details
We can start by using the midpoint formula which states that the midpoint of a line joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
So, the midpoint of the line joining (1-k, -4) and (2, k+1) is:
\begin{align*}
&\left(\frac{(1-k)+2}{2},\frac{(-4)+(k+1)}{2}\right) \\
&\Rightarrow \left(\frac{3-k}{2},\frac{k-3}{2}\right)
\end{align*}
We are given that the midpoint is (-k, k), so we can equate the x and y coordinates:
\begin{align*}
\frac{3-k}{2} &=-k \\
\frac{k-3}{2} &=k
\end{align*}
Solving for k, we get:
\begin{align*}
k &= 2 \\
\end{align*}
Therefore, the value of k is 2, which is the answer option labeled as "-2".