A triangle that has interior angles of $\displaystyle 30^{\circ}$ and $\displaystyle 90^{\circ}$ is necessarily a 30-60-90 triangle—a special right triangle. We can tell that the third angle about which we're not told anything has to be $\displaystyle 60^{\circ}$ because a triangle's interior angles always sum to $\displaystyle 180^{\circ}$, allowing us to solve for the third angle like so:

$\displaystyle a+b+c=180^{\circ}$

$\displaystyle 30^{\circ}+b+90^{\circ}=180^{\circ}$

$\displaystyle 120^{\circ}+b=180^{\circ}$

$\displaystyle (120^{\circ}+b)-120^{\circ}=(180^{\circ})-120^{\circ}$

$\displaystyle b=60^{\circ}$

Since we know this triangle is a 30-60-90 triangle, we can use the special ratios that always hold true for this triangle's sides and angles to figure out the lengths of its other sides. The following ratio holds true for all 30-60-90 triangles, where the side in a fraction with a given angle is the side opposite that angle.

$\displaystyle \frac{30^{\circ}}{x}=\frac{60^{\circ}}{x\sqrt3}=\frac{90^{\circ}}{2x}$

We're told that the hypotenuse of our triangle has a length of $\displaystyle 2\;\textup{in}$. The hypotenuse is the triangle's longest side, so it will be located directly across from its largest angle. In this case, that angle is $\displaystyle 90^{\circ}$. So, we need to set $\displaystyle 2x$ equivalent to $\displaystyle 2\;\textup{in}$ and solve for $\displaystyle x$.

$\displaystyle 2x=2$

$\displaystyle x=\frac{2}{2}=1$

As you can see, for this particular triangle, $\displaystyle x=1$. Using this information, we can now calculate the lengths of the other sides of the triangle. The side opposite the $\displaystyle 30^{\circ}$ angle will be equal to $\displaystyle x$ inches; since $\displaystyle x=1$, this side's length is $\displaystyle 1\;\textup{in}$. The side opposite the $\displaystyle 60^{\circ}$ angle will be equal to $\displaystyle x\sqrt3\;\textup{in}$. Substituting in $\displaystyle x=1$ into this expression, we find that this side has a length of $\displaystyle \sqrt{3}\;\textup{in}$.

Thus, the correct answer is $\displaystyle \sqrt3\;\textup{in}\;\textup{and}\;1\;\textup{in}$.

Since $\displaystyle \angle A$ is a right angle - that is, $\displaystyle m \angle A = 90$ - and $\displaystyle m \angle C = 30$, it follows that

$\displaystyle m \angle B = 180 - (m \angle A + m \angle C) = 180 - (90+30) = 60$,

making $\displaystyle \bigtriangleup ABC$ a 30-60-90 triangle. 

By the 30-60-90 Triangle Theorem, 

$\displaystyle AB = \frac{1}{2}(BC) = \frac{1}{2}(16) = 8$,

and

$\displaystyle AC = AB \sqrt{3} = 8\sqrt{3}$

Refer to the diagram below:

The area of a right triangle is equal to half the product of the lengths of its legs, so

$\displaystyle a = \frac{1}{2}(AB)(AC) = \frac{1}{2}(8)(8 \sqrt{3})= 32 \sqrt{3}$,

the correct response.