Question

based on the diagram, which expresses all possible lengths of segment ab?
options:
ab < 27 or ab > 81
ab = 25
27 < ab < 81
ab = 85
(diagram shows triangle abc with ac = 27, bc = 54)

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. For triangle \(ABC\) with sides \(AC = 27\), \(BC = 54\), and \(AB\) (the side we are interested in), we use the formula: \(|BC - AC| < AB < BC + AC\).

Step2: Calculate the bounds

First, find the difference of the two known sides: \(|54 - 27| = 27\). Then, find the sum of the two known sides: \(54 + 27 = 81\). So, applying the triangle inequality theorem, we get \(27 < AB < 81\).

Answer:

\(27 < AB < 81\) (the option with this inequality)