The diagram shows two identical triangular pieces of paper A and B. The side lengths of each triangle are in the proportion 3, 4 and 5. Each triangle is folded along a line through a vertex, so that the two sides meeting at this vertex coincide. The regions not covered by the folded parts have respective areas SA and SB. If SA+SB=39, find the area of the original triangular piece of paper A.
I understand that the triangles which cover each other, become congruent triangles. But, I am stuck here. I do not know how to proceed. Please advise.
The big triangle is 3-4-5
The yellow triangle is a right triangle. And it shares an angle with the big triangle, so it must be similar to the big triangle. Lets call that angle $\alpha$
The leg adjacent to $\alpha$ in $A = 4$
The leg adjacent to $\alpha$ in the yellow triangle $= 5-3 = 2$
Each side is half as long as the corresponding side in A.