Answer: D

Small Hint:

Let the side adjacent to the $60^\circ$ angle be $x$, so the third side is $90-x$

Big Hint:

Apply the law of cosines with included sides $80$ and $x$

Solution:

Let the side adjacent to the $60^\circ$ base angle be $x,$ and let the opposite side be $90-x.$ The law of cosines gives $$\begin{aligned} (90-x)^2 &=80^2+x^2\\ &\quad{}-2(80)(x)\cos60^\circ. \end{aligned}$$ Simplifying yields $8100-180x=6400-80x,$ so $x=17.$ The side lengths are $17,\ 73,$ and $80,$ and the shortest is $17.$

Thus, the correct answer is D.