Length of equal sides of an isosceles triangle is $50$ cm and area is $1200$ sq cm. The base is —

The altitude to the base splits the isosceles triangle into two right triangles, giving area $=\tfrac12 b\sqrt{50^2-(b/2)^2}=1200$. Testing $b=60$: height $=\sqrt{2500-900}=40$, so area $=\tfrac12(60)(40)=1200$. Thus the base is $60$ cm.