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 triangle inequality theorem states that for any triangle, the length of any side must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. Mathematically, for a triangle with sides \(a\), \(b\), and \(c\), \(|a - b| < c < a + b\).

Step2: Identify the given sides

In triangle \(ABC\), we are given \(AC = 27\) and \(BC = 54\). We need to find the possible length of \(AB\).

Step3: Apply the triangle inequality theorem

First, calculate the sum of the two given sides: \(27 + 54 = 81\).
Then, calculate the absolute difference of the two given sides: \(|54 - 27| = 27\).

According to the triangle inequality theorem, the length of \(AB\) must be greater than \(27\) and less than \(81\), i.e., \(27 < AB < 81\).

Answer:

\(27 < AB < 81\) (corresponding to the option "27 < AB < 81")