A geometry problem by Aaryan Maheshwari

Isosceles Triangles have 2 sides of equal length, so the possible sidelengths that we can form are $5+5+16=26$ or $5+16+16=37$ . But consider the scaled set-ups below: The sum of the two shorter lengths must be greater than that of the longest side. So we can rule out the possibility of $26$ , leaving the answer of $\boxed{37}$ .

If the triangle is isosceles, then two of its sides should be equal. Thus, there are two possible perimeters: $5+5+16 = 26\space units$ or $16+16+5 = 37\space units$ . But according to the triangle inequality , $26\space units$ cannot be a perimeter as $10<16$ , that leaves the only choice, which is $\fbox{37}$