A trapezoid is inscribed in a semicircle of radius r. What is the maximum perimeter and what is the length of the four sides when it does occur?

Right triangles ACB and AKC are similar. Consider the ratio of the shortest side to the hypotenuse in each to set up a proporton

Thus

The perimeter is 2r + x + 2s and so the perimeter function is given by

The value of the function in the interval from 0 < x < 2r can be explored with spreadsheet, graphing device, thinking, or other means (such as calculus).

Click here for a graph of the function when r = 10.