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# A triangle with integer sides

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A triangle with integer sides has perimeter 15. How many such non-congruent triangles are there? (A 4-5-6 triangle is considered congruent to a 6-5-4 triangle because we can reflect and rotate the triangles until they match up.)

Oct 28, 2020

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Let the sides of the triangle be \$a\$, \$b\$, and \$c.\$

Then by the triangle inequality,

\$\$a+b>c\$\$ \$\$b+c>a\$\$ \$\$c+a>b\$\$

We also have that \$a+b+c=15.\$

The longest side of this triangle must be shorter than \$15/2=7.5\$. However, we have to round this up to get an integer, so the longest side must be shorter than \$8.\$ This is because the remaning sides must sum to less than \$8,\$ or \$7,\$ and we won't have a triangle!

Now, to focus on how short the longest side might be, we know that if \$a=b=c\$ then all of the sides must be the longest side. This is possible if we have side lengths of \$5, 5, 5.\$ However, if the longest side is \$4,\$ the remaining sides must sum to \$15-4=11.\$ We must have more than a 4 for the longest side, as \$4+4+4 = 12.\$ Let the longest side be l. So, \$5\le l \le 7.\$

Listing out the possibilities with 5 as the longest side, \$\$(5,5,5).\$\$ With 6, \$\$(6,3,6)~(6,4,5)\$\$ And with 7, \$\$(7,1,7) ~ (7,2,6) ~ (7,3,5) ~ (7,4,4)\$\$  we have a grand total of \$\$\boxed{7}\$\$ possibilities for non-congruent triangles.

Oct 28, 2020
edited by Pangolin14  Oct 28, 2020
edited by Pangolin14  Oct 28, 2020