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If $2$ altitudes of a triangle with integer side lengths are $9$ and $40$ units in length, then find the minimum possible perimeter of the triangle

Since the altitude is the shortest distance from a vertex to the opposite side, I got $a>9$ and $b>40$, where $a$ and $b$ are two of the sides of the triangle. For minimum perimeter I took $a_{min}=10,b=41_{min}$ which gave me $31<c<51 \implies c_{min}=32$. Hence I concluded that the minimum perimeter was $83$. However, the answer given is $90$. I suspect this is the right angled triangle of $(9,40,41)$ but I couldn't see why that would be the one with the smallest perimeter.

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  • $\begingroup$ A) Do you understand why 83 is not attainable? B) Are you making the assumption that the side lengths must be integers, or is that given? $\endgroup$ – Calvin Lin Nov 24 '20 at 3:32
  • $\begingroup$ @Calvin Lin A) no, that's exactly what I don't understand B) yes, I forgot to mention that in the question; edited now $\endgroup$ – l1mbo Nov 24 '20 at 3:33
  • $\begingroup$ A) Actually $ a \geq 9$. B) In order for $ a_{min} = 9$, what other conditions must hold? $\endgroup$ – Calvin Lin Nov 24 '20 at 3:34
  • $\begingroup$ @Calvin Lin I understood now, fixing two sides and having a fixed altitude fixes the third side, thx for help $\endgroup$ – l1mbo Nov 24 '20 at 3:44
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I want to add something to Calvin Lin's comment to make it more rigorous.

By equating area we have $$40a=9b$$ Now since $a,b$ are integers and $\gcd(9,40)=1$ we must have $a=9k,b=40k$ for some natural $k$. If $k=1$ we see that the condition is satisfied and $a+b+c=90$.If $k\ge 2$ then we have $a+b+c>a+b=49k\ge 98$ hance minimum of $a+b+c$ is indeed $90$.

Note:Just because $a_{min}=9$ we cant say that the minimum value of $a+b+c$ occurs when this occurs(since a,b,c are not completely independent variables)

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    $\begingroup$ Thanks that was really great! I just made a rough figure, got some intuition and moved on, big mistake $\endgroup$ – l1mbo Nov 24 '20 at 5:32
  • $\begingroup$ +1 (I wasn't giving a solution via the comment, but just trying to guide OP to do more thinking.) $\endgroup$ – Calvin Lin Nov 24 '20 at 16:51
  • $\begingroup$ @CalvinLin yes i guessed that,i wrote this answer as i thought the OP had got a misconception that arbitrarily we can take a+b+c to be min when 'a' is minimum' as posted above $\endgroup$ – Albus Dumbledore Nov 24 '20 at 16:55

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