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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 21, 2022
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Use Heron's Formula to find the area

s = the semi-perimeter  =   ( 8 + 12 + 14)  / 2  =    17

 

Area =  sqrt  [ 17 (17- 8) (17 - 12) ( 17 - 14) ]    =   3sqrt (255)

 

The longest altitudes will be drawn to the two shortest sides

 

So

 

3sqrt (255)   =(1/2)(8) * altitude 1              3sqrt (255)  =(1/2)(12) * altitude 2

(3/4)sqrt (255) = altitude 1                         (1/2)sqrt (255)  = altitude 2

 

Sum of altitudes   =    [3/4 + 1/2]sqrt (255)  =  1.5 sqrt (255)

 

cool cool cool

 Apr 21, 2022

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