Question

based on the diagram, which expresses all possible lengths of segment ab?
ab = 25
27 < ab < 81
ab = 85
ab < 27 or ab > 81

Explanation:

Step1: Apply triangle inequality theorem

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side. Let \(AC = 27\), \(BC=64\), and \(AB=x\). Then \(|AC - BC|\lt AB\lt AC + BC\).

Step2: Calculate the range

First, calculate \(AC + BC=27 + 64=91\). Second, calculate \(|AC - BC|=|27-64| = 37\). So \(37\lt AB\lt91\).

Answer:

\(27\lt AB\lt91\) (assuming there was a typo in the options and it should be \(37\lt AB\lt91\) based on the formula, but among the given options, if we consider the closest correct - like form from the triangle - inequality - sum \(27 + 64=91\) and difference \(|64 - 27| = 37\), perhaps a mis - write in the problem's options, but the intended answer from the structure of options is \(27\lt AB\lt91\))