Question 1125927
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The standard strategy for solving such problems is to calculate the area of the triangle using the Heron's formula;

then to divide the area by the half of the base.



The perimeter of the triangle is  13 + 37 + 40 = 90 cm;  hence, the semi-perimeter is  90/2 = 45 cm.



By the Heron's formula, area of the triangle is


    A = {{{sqrt(45*(45-40)*(45-37)*(45-13))}}} = {{{sqrt(45*5*8*32)}}} = {{{sqrt(9*5*5*2^3*2^5)}}} = {{{3*5*2^4}}} = 3*5*16 = 240 cm^2.



Then the length of the altitude to the longest side  = {{{240/((40/2))}}} = {{{240/20}}} = 12 cm.
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Solved.