Question

points a, b, and c, form a triangle. the distance between point a and point b is 15 yards. the distance between point b and point c is 25 yards. pete walks directly from point a to point c, without passing through point b. what is the direct distance from a to c?
how far would pete walk if he went from a to b to c?
the direct distance from a to c is more than
the inequality w < represents the distance, w, that pete might save by taking the direct path.

Explanation:

Step1: Apply the triangle inequality theorem

The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let \(AB = 15\) yards (\(a\)), \(BC = 25\) yards (\(b\)), and \(AC=w\) yards (\(c\)).

Step2: Calculate the direct distance from \(A\) to \(C\)

By the triangle inequality \(|a - b|\(|15 - 25|\(10

Step3: Find the distance if Pete goes from \(A\) to \(B\) to \(C\)

The distance from \(A\) to \(B\) to \(C\) is \(AB+BC\). Since \(AB = 15\) and \(BC = 25\), \(AB + BC=15 + 25=40\) yards.

Step4: Find when the direct distance from \(A\) to \(C\) is more than a certain value

We know \(w>10\) (from the triangle - inequality \(|a - b|

Answer:

The direct distance from \(A\) to \(C\) is more than \(10\) yards. Pete would walk \(40\) yards if he went from \(A\) to \(B\) to \(C\). The inequality \(w < 40\) represents the distance \(w\) that Pete might save by taking the direct path.