An isosceles triangle has two of its sides equal to 5 and 6. Find its perimeter.

Step-by-step solution

1. 1

   Recognise that the problem is deliberately ambiguous. "Two sides are equal" and "two of the sides measure $5$ and $6$" do not say which side is repeated. The third side must equal one of the two given lengths, so there are two candidate triangles and both must be examined.

2. 2

   Case 1: the repeated side is 5. The sides are $5,5,6$. Check the triangle inequality — the two shorter sides must exceed the longest:

   $5+5=10>6\quad\checkmark$

   So this triangle exists, and its perimeter is

   $5+5+6=16$

3. 3

   Case 2: the repeated side is 6. The sides are $6,6,5$. Again check:

   $6+5=11>6\quad\checkmark$

   This triangle also exists, with perimeter

   $6+6+5=17$

4. 4

   Report both answers. Since both configurations are valid, the perimeter is not unique:

   $\boxed{16\ \text{or}\ 17}$

5. 5

   Note when a case would have to be rejected. The triangle inequality is not a formality here — with given lengths $2$ and $5$, the case $2,2,5$ fails because $2+2=4<5$, leaving only $5,5,2$ and a single perimeter of $12$. Always run the test on both cases before deciding how many answers there are.