Problem:
In an isosceles trapezoid, the parallel bases have lengths and , and the altitude to these bases has length . The perimeter of the trapezoid can be written in the form , where and are positive integers. Find .
Solution:
Let be one of the vertices of the smaller base, let be the foot of the altitude from to the larger base, and let be the vertex of the larger base closer to . Because the trapezoid is isosceles, it follows that . Note that ; hence right has sides in the ratio of , and thus . The perimeter of the trapezoid is therefore . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions