Quite easy for AIME, see if you can figure it out.

Square AIME has sides of length 10 units. Isosceles triangle GEM has base EM, and the area common to triangle GEM and square AIME is 80 square units. Find the length of the altitude to EM in GEM.

meh wish this forum had asy.

Let the intersection point of AE and GE be X, and let the intersection point of AE and GM be Y.

(XY+EM)*EM/2=80, (XY+10)*10/2=80, XY=6

Let the foot of perpendicular of G to AE be Z.

proportion - triangle GXY and triangle GEM

GZ : 6 = (GZ+10) : 10, 6GZ+60=10GZ, GZ=15.

therefore, altitude to EM in GEM is 10+15=25.

answer : 25