Since triangles TIP and TOP are isosceles, this means that sides TI and TP are equal, and sides TO and TP are equal.
Given: TI = 5, PI = 7, PO = 11.
We can see that triangle TIP is not congruent to triangle TOP since the sides are not the same length. Therefore, we have two separate cases to consider:
**Case 1: Triangle TIP**
In triangle TIP, sides TI and TP are equal, so TP = 5. Since TI + PI > TP, we have 5 + 7 > 5, which is true. So, triangle TIP is valid.
**Case 2: Triangle TOP**
In triangle TOP, sides TO and TP are equal, so TP = TO. Since TO + PO > TP, we have TO + 11 > TO, which implies 11 > 0, which is true. So, triangle TOP is valid.
Considering both cases, we see that there are no restrictions on the length of TO. It can be any positive real number. Thus, all possible lengths of TO are **all positive real numbers**.