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Olympiad workbookIOQMProperties of Triangles

If 15 and 9 are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

Correct answer

90

Step-by-step solution

Area of BDG aligned & S = 11+x 2 & aligned Area = & 11+x 2 ( 11+x 2 -x ) ( 11+x 2 -5 ) ( 11+x 2 -6 ) & = 11+x 2 ( 11-x 2 ) ( x+1 2 ) ( x-1 2 ) & = (121-x^2 ) (x^2-1 ) 16 aligned & Let y = (121- x ^2 ) ( x ^2-1 ) 16 & For max area dy dx =0 & -2 x ( x ^2-1 )+ (121- x ^2 ) 2 x =0 & - x ^2+1+121- x ^2=0 aligned aligned & 2 x^2=122 & x^2=61 & Area of BDG = (121-61)(61-1) 16 & = 60 60 16 = 60 4 =15 aligned Area of ABC =6 15=90

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