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 \(a = 90\) yards, \(b = 60\) yards, and \(c\) be the length of the third hallway.

Step2: Calculate the lower bound

First, find the difference of the two known sides: \(|90 - 60| = 30\) yards. So, \(c\) must be greater than 30 yards.

Step3: Calculate the upper bound

Next, find the sum of the two known sides: \(90 + 60 = 150\) yards. So, \(c\) must be less than 150 yards.

Answer:

between 30 yards and 150 yards