How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides 8, 12, and 14?
+1164 ANSWER BY CPhill:
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)
