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: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle 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 the sides of the triangle be \(a = 27\), \(b = 54\), and \(c=AB\).

Step2: Calculate the lower - bound

\(b - a\lt c\), so \(54 - 27\lt AB\), which simplifies to \(27\lt AB\).

Step3: Calculate the upper - bound

\(a + b>c\), so \(27+54 > AB\), which simplifies to \(AB < 81\).

Answer:

\(27 < AB < 81\)