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Excellent job, Phill !!!
Sorry to say, but this is wrong!!
Semi-perimeter is 18.5
Should be: A = sqrt [18.5 (12.5)(5.5)(0.5)] = 25.21780125
btw, the right triangle with sides of 6 and 13 has a larger area.
Two sides of a triangle are 6 and 13. What is the largest possible area of the triangle?
The largest area of the triangle with the sides of 6 units and 13 units is 39 units squared.
I drew the rectangle with the sides of 5 and 4, and it didn't work.
Eventually, I decreased the shorter side to 3 units, and it worked like a charm.
See my diagram regarding the lengths of sides of a blue region.
Let the side lengths of the rectangle BCDF be 3 cm and 5 cm
AB = 3.333333333 cm
EN = 1.2 cm EM = 2 cm
Area of the blue region is 5.5 cm²
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M is a midpoint of AB
Angles MAC and ABO are equal
tan(ABO) = 3 / 12 = 14.03624347º
AM = {sqrt[(AO)² + (BO)²]} /2 = 6.184658438
r = AM / cos(MAC) = 6.375
