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How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides 8, 12,  and  14?

 Apr 30, 2022
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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.

 May 2, 2022

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