7. The three vertices of a right triangle with an angle of $54^{\circ}$ are also vertices of a regular polygon having...
(A) 12 sides
(B) 25 sides
(C) 30 sides
(D) 24 sides
(E) 16 sides
(7) The correct answer is $(C)$.

First, it can be observed that, since a regular polygon can be inscribed in a circle, the presence of a right angle implies that the hypotenuse of the triangle must be a diameter of that circle. Therefore, the regular polygon must have an even number of sides.

The other two angles of the triangle measure $54^{\circ}$ and $36^{\circ}$: considering the circumcircle of the polygon, the central angles corresponding to the arcs subtended by these angles measure $108^{\circ}$ and $72^{\circ}$. These arcs are divided into equal parts by the vertices of the polygon, so the central angles corresponding to the sides of the regular polygon must have a measure that is a submultiple of $108^{\circ}$ and $72^{\circ}$, i.e., a submultiple of their GCD, which is $36^{\circ}$.

Therefore, the number of sides must be a multiple of 10. Among the given answers, only 30 is.

Question proposed by Maria Chiara Ricciuti.