Here is the step-by-step solution to determine the possible lengths of segment AB in the given triangle ABC:
Let's break this down:
- We are given:
AC = 27
CB = 54
- We want to find: Possible lengths of AB
- To do this, we can utilize the triangle inequality theorem, which states:
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Applying this:
AC + CB > AB
27 + 54 > AB
81 > AB
- This means AB must be less than 81.
- Additionally, we know:
AB + AC > CB
AB + 27 > 54
AB > 27
- Therefore, AB must be greater than 27.
- Bringing this together:
27 < AB < 81
So the possible lengths of segment AB are:
27 < AB < 81
Therefore, the answer is option B.
By methodically applying the triangle inequality theorem, we have determined the only possible lengths for side AB given the constraints.