Suppose triangle TIP and triangle TOP isosceles triangles. Also suppose that TI=5, PI=7 and PO=11.

What is the sum of all possible lengths for TP?

Since TIP and TOP are isosceles triangles, all their corresponding sides must be equal.

Therefore, TP must be equal to PO, the longest side, which is 11.

So the sum of all possible lengths for TP is simply 11.