Question

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 value represent the possible distance, d, in miles, between lincoln, ne, the third city?

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for a 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

The lower - bound is \(|500 - 200|\).

$$|500 - 200|=300$$

Step3: Calculate the upper - bound

The upper - bound is \(500 + 200\).

$$500+200 = 700$$

Answer:

\(300\lt d\lt700\)