Calculate 3rd Side of Triangle When Only 2 Sides Known and Angles Not Known?

Question

I have a polygon with 4 sides (going clockwise) of 10,12,15,9. I do not know any angles, none of the angles are right angles, the sides are not parallel. If I draw a line diagonally from one corner to another, I now have two triangles within my polygon in which I know the lengths of the two sides. I am unable to physically measure that line I drew or anything else. Nor am I able to measure any angles.

What is the formula, equation or process I would use to determine the length of that diagonal line (the third side of the triangle). Then, how would I calculate the area of each triangle?

Answer 1

Given these data, the length of one diagonal can be anything between 6 and 22, and the length of the other anything between 3 and 19.  Those limits follow from the triangle inequality, which says that the length of a side of a triangle is less than the sum of the two other sides.  If the length of one diagonal is known, there can be up to four possible lengths for the other diagonal.