The lengths of two sides of an isosceles triangle are $15$ and $22$. Which of the following are the **possible** values of the perimeter? (An isosceles triangle has at least two equal sides.)

An isosceles triangle has two equal sides. Two cases: **Case 1**: the equal sides are both $15$. Third side is $22$. Triangle inequality: $15 + 15 = 30 > 22$ ✓. Perimeter $= 15 + 15 + 22 = 52$. **Case 2**: the equal sides are both $22$. Third side is $15$. Triangle inequality: $22 + 22 = 44 > 15$ ✓ and $22 + 15 = 37 > 22$ ✓. Perimeter $= 22 + 22 + 15 = 59$. Both cases satisfy the triangle inequality, so both perimeters are possible. - Traps A and B each pick only one case. - Trap D wrongly rules out the configuration. Key: in a side-length problem with isosceles, always check **both** assignments of the equal pair.