What's the midpoint of $[a,b)$? While the midpoint of either $[a,b]$ or $(a,b)$ is $\dfrac{a+b}{2}$, what's the midpoint of $[a,b)$ if it has been defined at all?
It depends on how you want to define the so called "mid point" of an interval. One can simply define the mid point of an interval to be $\frac{a+b}2$ for any interval of the form $(a,b), (a,b], [a,b)$ or $[a,b]$. This definition would serve most general purposes quite well since a single point has measure zero. 

However, suppose you want the define the mid point to have the property such that 


  for an interval $I$, $m$ is called the midpoint of $I$ if for any $x\in I$ there exist $x'\in I$ such that $x-m=m-x'$


then it's provable that $[a,b)$ has no mid point. (Hint: if such $m$ exists then $I$ must be of the form $(m-d,m+d)$ or $[m-d,m+d]$)
The standard definition of the midpoint is, "The point on a line segment dividing it into two segments of equal length" or the equivalent. As a result, the midpoint is found by the formula $\frac{a + b}{2}$ for all types of intervals. 


https://en.wikipedia.org/wiki/Midpoint
http://mathworld.wolfram.com/Midpoint.html