Question

fionas school is in the shape of a triangle, with three hallways connecting the corners. fiona enters the building and walks 90 yards through the first hallway to the first corner. then, she walks 60 yards through the second hallway to the second corner. what are the possible lengths of the third hallway?
○ between 30 yards and 60 yards
○ between 30 yards and 90 yards
○ between 30 yards and 120 yards
○ between 30 yards and 150 yards

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b| < c < a + b\). Let the lengths of the two known sides be \(a = 90\) yards and \(b = 60\) yards, and the unknown side be \(c\).

Step2: Calculate the lower bound

First, find the lower bound of \(c\) by calculating \(|a - b|\).
\(|90 - 60| = 30\) yards. So, \(c>30\) yards.

Step3: Calculate the upper bound

Next, find the upper bound of \(c\) by calculating \(a + b\).
\(90 + 60 = 150\) yards. So, \(c < 150\) yards.

Answer:

between 30 yards and 150 yards