# Greatest possible perimeter [closed]

What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 cm? Explain the procedure

• If we as a guess take the triad (12,13,5) of perimeter 30 we scale to multiply by 12/5 or, 12/5* 30 = 72. Commented Aug 13, 2016 at 14:04

Obviously 12 must be the base for maximum perimeter

By Pythagoras theorem

$144 = a^2-c^2$

So $144= (a-c)(a+c)$

For maximum perimeter we need $a+c$ maximum. $a+c =144$ dosent give integer solutions for a and c. The next largest choice on factoring $144$ is $72$ This gives integer solutions. Hence perimeter is $84$

Note: Here $a$ is the hypotenuese and $c$ is the other base.

You have to think of the Pythagorean Theorem.

Let $$a,b$$ be the legs of the triangle and $$c$$ be the hypoteneuse. Then, of course:

$$a^2=c^2-b^2=(c+b)(c-b)$$

Here, you have $$a=12$$ so $$a^2=144$$. So the whole numbers $$b$$ and $$c$$ must be selected so that their sum times their difference equals $$144$$.

For any two whole numbers the sum and difference are both even or both odd. To make the product $$144$$ you have to make the sum and difference both even, so you choose a pair of even numbers with that product. Let's try $$18$$ and $$8$$. The larger of these two factors is $$c+b=18$$ and the smaller one is $$c-b=8$$. Then $$c$$ is the average of these two numbers, thus $$c=13$$, and that leaves $$b=5$$. Note that even though the sum and difference must be even each individual side can be odd (if so, both are odd).

For the triangle above, then, the perimeter is $$a+b+c=12+5+13=30$$, which is not the maximum. That's because I deliberately made a poor choice of factors for $$144$$. Your job is to find a pair of even factors of $$144$$ so that the larger one, being the sum of the other two sides, is as big as you can get it. Then you get the maximum perimeter you want.

• Please explain to me what was wrong. I cannot figure out myself why I was downvoted. Commented Aug 13, 2016 at 12:43
• I find your answer is better than the other with an upvote. Commented Aug 13, 2016 at 12:48
• @N.S.JOHN: Your answer is good, dear friend, but good is not optimal. (neither Oscar Lanzi's answer!) Commented Aug 14, 2016 at 15:37
• @Piquito will keep that in mind Commented Aug 14, 2016 at 15:38