SOLUTION: What is the length of the shortest altitude of a triangle with the sides of 13, 14, and 15?

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Question 557966: What is the length of the shortest altitude of a triangle with the sides of 13, 14, and 15?
Found 2 solutions by richard1234, Edwin McCravy:
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The shortest altitude will be the one intersecting the side with length 15 (this is because the altitude is inversely proportional to the base). The area of a 13-14-15 triangle is 84 (useless fact, but it appears on numerous AMC/AIME problems. You could also use Heron's formula).

We now have 84 = 15a/2 where a is the altitude. Solving, we get 168 = 15a, a = 168/15 = 56/5.

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

The other tutor's answer is correct, but maybe you're studying the 
law of cosines instead of Heron's formula.  And maybe you haven't
proved that the shortest altitude is the one to the longest side.
(His "inverse proportional" statement is incorrect).

Maybe this is the approach you should take.
 

 
Let's draw the three altitudes:
  

 
The law of cosines for the cos(A) is

cos%28A%29 = %28b%5E2%2Bc%5E2-a%5E2%29%2F%282bc%29 = %2813%5E2%2B14%5E2-15%5E2%29%2F%282%2A13%2A14%29 = 140%2F264 = 5%2F13 
 
h%5B1%5D%2Fb = sin%28A%29 = sqrt%281-cos%5E2%28A%29%29 = sqrt%281-%285%2F13%29%5E2%29 = sqrt%281-25%2F169%29 = sqrt%28169%2F169-25%2F169%29 = sqrt%28144%2F169%29 = 12%2F13
 
h%5B1%5D%2Fb = 12%2F13
h%5B1%5D%2F13 = 12%2F13

Multiply both sides by 13:

h%5B1%5D+=+12
 
The law of cosines for cos(B)

cos%28B%29 = %28a%5E2%2Bc%5E2-b%5E2%29%2F%282ac%29 = %2815%5E2%2B14%5E2-13%5E2%29%2F%282%2A15%2A14%29 = 252%2F420 = 3%2F5

h%5B2%5D%2Fc = sin%28B%29 = sqrt%281-cos%5E2%28B%29%29 = sqrt%281-%283%2F5%29%5E2%29 = sqrt%281-9%2F25%29 = sqrt%2825%2F25-9%2F25%29 = sqrt%2816%2F25%29 = 4%2F5
 
h%5B2%5D%2Fc = 4%2F5
h%5B2%5D%2F14 = 4%2F5
Cross multiply
5h%5B2%5D+=+56
h%5B2%5D+=56%2F5
h%5B2%5D=11.2

h%5B3%5D%2Fc = sin%28A%29 which we have already calculated as 12%2F13

h%5B3%5D%2Fc = 12%2F13

h%5B3%5D%2F14 = 12%2F13
Cross multiply
13h%5B3%5D+=+168
h%5B3%5D+=168%2F13
h%5B3%5D=12.92307692


So h%5B2%5D=11.2 is the shortest altitude.

Incidentally, if you draw all three altitudes, they all three
intersect at the same point, called the orthocenter.


Edwin