1. **Problem statement:** Find the length of side $x$ in a right triangle where the angles are $30^\circ$, $60^\circ$, and $90^\circ$, with the side opposite the $30^\circ$ angle given as 7. 2. **Formula and rules:** In a $30^\circ$-$60^\circ$-$90^\circ$ triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. - The side opposite $30^\circ$ is the shortest side, call it $a$. - The side opposite $60^\circ$ is $a\sqrt{3}$. - The hypotenuse is $2a$. 3. **Given:** The side opposite $30^\circ$ is 7, so $a = 7$. 4. **Find:** The side opposite $60^\circ$, which is $x = a\sqrt{3} = 7\sqrt{3}$. 5. **Simplify:** The expression $7\sqrt{3}$ is already in simplest radical form with a rational denominator. **Final answer:** $$x = 7\sqrt{3}$$