The point is on the bisector of the angle ∠ A.

Because of isosceles, the angular bisector AO is perpendicular to BC, MC=0.5* 12=6 (three lines are one) and OM⊥BC.

So an O M three-point line. AM⊥BC

AM=8 of Pythagoras' law

So the triangle area is 8* 12*0.5=48.

If the small line segment is set to x (10x+10x+12x) * 0.5 = 48, X=3.

Right? Ha ha laugh