Altitude to Longest Side of triangle w/ sides 9, 12, 15

[The Paradox Obsessed]

Question: Triangle ABC has sides of 9, 12, and 15. What is the length of the altitude to the longest side?
Work:
So far I have found that not only is it a right triangle by the Pythagoran Theorem (9^2+12^2=15^2) is a Pythagorian Triple (3*3=9, 3*4=12, 3*5=15).
I am having issues, however, finding the actual altitude. It would be the altitude to the longest side, which is the hypotenuse of this particular triangle. Please can I get some help?

 

[Denis]

Re: Altitude to the Longest Side

Hint:
Triangle's area = 9 * 12 / 2 = 54
(longest side) * (altitude) / 2 = 54

 

[The Paradox Obsessed]

Re: Altitude to the Longest Side

Thank you so much. That makes a lot more sense.

 

[soroban]

Re: Altitude to the Longest Side

Hello, The Paradox Obsessed!

Another approach . . .

Triangle ABC has sides of 9, 12, and 15.
What is the length of the altitude to the longest side?

            A
            *
           *|  *
          * |     *   12
       9 *  |4a      *
        *   |           *
       *@   |              *
    B *  *  *  *  *  *  *  *  * C
        3a  D

We find that: .\(\displaystyle \Delta DBA \sim \Delta ABC\)

The sides of \(\displaystyle \Delta DBA\) are also in the ratio \(\displaystyle 3:4:5.\)

\(\displaystyle \text{We have: }\;BD \,=\, 3a,\;AD \,=\, 4a,\:\text{ and }\:AB \,=\, 5a \,=\, 9 \quad\Rightarrow\quad a \,=\, \tfrac{9}{5}\)

\(\displaystyle \text{Therefore: }\;\text{altitude} \:=\:AD \:=\:4a \:=\:4\left(\tfrac{9}{5}\right) \:=\:\frac{36}{5}\)