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

B. 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

1)
`AC+BC > AB`

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

2)
`27+AB > 54`

`AB > 54-27\\ AB > 27`

3)
`54+AB > 27`

`AB > 27-54\\ AB > -27`

therefore

the answer is

`27 < AB < 81`