Low Variance Measure

Question

We have two rods of lengths $a$ and $b$. However, we only have a noisy measure of each rod. The independent random observed lengths $A$ and $B$ of the two rods are such that $\ev{A} = a, \ev{B} = b,$ and $\var{A} = \var{B} = \sigma^2$. You measure the rods in the following way: Take $U = A+B$ and $V = A-B$, and define $\hat{a} = \frac{U+V}{2}$. When $a = 12, b = 710$ and $\sigma^2 = 16$, report $\var{\hat{a}}$.