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

Answer 1

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`

Answer 2

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