6). KS=?

In triangle KPS, angle P is 90 degrees and angle S is 45 degrees. Therefore, angle K is also 45 degrees, making it an isosceles right triangle. PO is the altitude to the hypotenuse. Since KO = 8, PS = KP. In triangle KPS, tan(45) = KP/PS, which is 1. Also, in right triangle POS, angle POS = 90, angle S = 45, so angle OPS = 45. Thus, triangle POS is isosceles with PO = OS = 8. Since triangle KPS is isosceles right triangle, PO is also the median, so KS = 2 * PO = 2 * 8 = 16.