Question

If the perimeter of a triangle with integral sides is $17$ units, then the number of such triangles is:

• A. $6$
• B. $8$
• C. $10$
• D. $12$

Answer 1

Answer: (B)

$8$

For the number of triangles with Integer sides for a given perimeter, we have the below formulae:

If the perimeter $p$ is even then, the total number of such triangles is $\frac{\left[p^2\right]}{48}$

If the perimeter $p$ is odd then, the total number of such triangles is $\frac{\left[\right(p+3{)}^2]}{48}$

Where $\left[x\right]$ is the greatest integer less than or equal to $x$

Given: $p=17$

Total number of such triangles is

$\frac{\left[\right(p+3{)}^2]}{48}$

$=\frac{\left[\right(17+3{)}^2]}{48}$

$=\frac{400}{48}=8.3$

$\therefore$ No. of such triangles $=8$.