Question

Based on the diagram, which expresses all possible lengths of segment AB? AB = 25 27 < AB < 81 AB = 85 AB< 27 or AB &gt; 81

Answer 1

The answer

according to the figure, we can solve this problem only by applying sines rule:
that is 
sinA/a  = sinB/b = sinC /c

As we observe,  sin A /54  = sin B/27 = sin C/ c, and c = AB

besides, sinC > sinB > sinA  , so the only answer possible is 
27 < AB < 81 

Answer 2

we know that

The sum of the lengths of any two sides of a triangle is greater than the length of the third side (Triangle Inequality Theorem)

So

see the attached figure to better understand the problem

1) $AC+BC > AB$

$27+54 > AB\\ 81 > AB\\ AB < 81$

2) $27+AB > 54$

$27+AB > 54\\ AB > 54-27\\ AB > 27$

3) $54+AB > 27$

$54+AB > 27\\ AB > 27-54\\ AB > -27$

therefore

the answer is the option

$27 < AB < 81$