Let's start by making a sketch:
To figure out the perimeter, we will need all sides.
Since this is a right triangle, we can use trigonometric ratios to find them.
We know the leg adjacent to the 30° angle, so we can use tangent of 30° to find the other leg and either cosine or sine of 30° to find the hypotenuse afterwards.
[tex]\begin{gathered} \tan 30\degree=\frac{\text{opposite leg}}{\text{adjacent leg}}=\frac{BC}{AC} \\ BC=AC\cdot\tan 30\degree \end{gathered}[/tex]
Tanget of 30° degree is known to be:
[tex]\tan 30\degree=\frac{\sqrt[]{3}}{3}[/tex]
So:
[tex]BC=AC\cdot\tan 30\degree=7\sqrt[]{2}\cdot\frac{\sqrt[]{3}}{3}=\frac{7\sqrt[]{6}}{3}[/tex]
And the value for cosine of 30° is also known:
[tex]\cos 30\degree=\frac{\sqrt[]{3}}{2}[/tex]
So:
[tex]\begin{gathered} \cos 30\degree=\frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{AC}{AB} \\ AB=\frac{AC}{\cos30\degree}=\frac{7\sqrt[]{2}}{\frac{\sqrt[]{3}}{2}}=\frac{14\sqrt[]{2}}{\sqrt[]{3}}=\frac{14\sqrt[]{2}\sqrt[]{3}}{3}=\frac{14\sqrt[]{6}}{3} \end{gathered}[/tex]
So, the perimeter will be:
[tex]\begin{gathered} P=AB+BC+AC \\ P=\frac{14\sqrt[]{6}}{3}+\frac{7\sqrt[]{6}}{3}+7\sqrt[]{2} \\ P=\frac{21\sqrt[]{6}}{3}+7\sqrt[]{2} \\ P=7\sqrt[]{6}+7\sqrt[]{2} \end{gathered}[/tex]
Which corresponds to the third alternative.
