A string around a circle is pulled up by a thumbtack, $T$, as shown below. The string is tangent to the circle with $AT = TB$. Given that the circle has a radius of $r$, complete the following problems.

1. Find an expression for the length of the string if $AT = r$.

   Answer (a):

   $1.5πr + 2r ≈ 6.712r$

2. How long would the string have to be if $\overarc { A B }$ is $120°$?

   Answer (b):

   $\frac{4\pi r}{3}+2r\sqrt3\approx 7.653\textit{r}$

3. Find $m∠ATB$.

   Answer (c):

   $60°$