For a body to be in equilibrium under three forces, the resultant of any two forces must be equal in magnitude and opposite in direction to the third force: $\vec{F}_1 + \vec{F}_2 = -\vec{F}_3 \implies |\vec{F}_1 + \vec{F}_2| = |\vec{F}_3|$. Since the angle between $\vec{F}_1$ ($5\text{ N}$) and $\vec{F}_2$ ($12\text{ N}$) is $90^\circ$, their resultant is $R = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\text{ N}$. Thus, the third force must have a magnitude of $13\text{ N}$ to keep the system in equilibrium. Other options represent algebraic sum ($17\text{ N}$) or difference ($7\text{ N}$), which are only valid if the angle between them is $0^\circ$ or $180^\circ$, respectively.