Question

determining possible side lengths
the distance between lincoln, ne, and
boulder, co, is about 500 miles. the distance
between boulder, co, and a third city is 200
miles.
assuming the three cities make a triangle on the
map, which values represent the possible
distance, d, in miles, between lincoln, ne, and
the third city?
<d<

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 500\) (distance between Lincoln, NE and Boulder, CO) and \(b=200\) (distance between Boulder, CO and the third city).

Step2: Calculate the lower - bound

First, find \(|a - b|\): \(|500 - 200|=300\).

Step3: Calculate the upper - bound

Then, find \(a + b\): \(500+200 = 700\).

Answer:

\(300\lt d\lt700\)