How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides 8, 12, and 14?
We calculate the area by Heron's formula,
s = (8 + 12 + 14)/2 = 17
Area = sqrt(s(s - 8)(s - 12)(s - 14)) = 3 sqrt(255)
Altitude length corresponding to a side x is 2(area)/x.
Use the shortest side lengths as x, and you will get the longest altitude in result. You can substitute area = 3 sqrt(255) and x = 8, x = 12 to get the length of the two longest altitudes.