Suppose $TIP$ and $TOP$ are isosceles triangles. Also suppose that $TI=5,$ $PI=7,$ and $PO=11$. What is the sum of all possible lengths for $TO$?

18

Step-by-step explanation:

If TP = 7, then TO can be 7 or 11 to make the triangle isosceles.

If TP = 5, then TO can 5 or 11

Stop right there, though! Your logic is correct, but TO cannot be 5! Why? Triangles have certain properties that they must adhere to for a triangle to exist. One such theorem is called the triangle inequality theorem. It states that the sum of the lengths of two different sides is greater than the remaining third side.

The other side length possibilities adhere to this restriction, thankfully. Now, add 7 and 11.  7+11=18