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
(diagram: triangle abc with ac = 27, cb = 54)

Explanation:

Step1: Recall Triangle Inequality Theorem

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

Step2: Identify the given sides

In triangle \(ABC\), we know two sides: \(AC = 27\) and \(BC = 54\). Let \(AB = c\), \(AC = a = 27\), and \(BC = b = 54\).

Step3: Apply the triangle inequality

First, calculate the sum of the two known sides: \(a + b = 27 + 54 = 81\).
Then, calculate the absolute difference of the two known sides: \(|b - a| = |54 - 27| = 27\).
By the triangle inequality theorem, \(27 < AB < 81\).

Answer:

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