Question

determining the difference in distance
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
yards.
the inequality w < represents the distance, w, that pete might save by taking the direct path.

Explanation:

Step1: Calculate the distance from A to B to C

The distance from A to B is 15 yards and from B to C is 25 yards.
$$15 + 25$$

Step2: Apply the triangle - inequality theorem

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let \(AB = 15\), \(BC=25\), and \(AC=x\). Then \(x>BC - AB\).
$$x>25 - 15$$

Step3: Find the maximum value of the saved distance

The saved distance \(w=(AB + BC)-AC\). Since \(AC>10\) (from step 2) and \(AC10\). The maximum value of \(w\) is when \(AC\) approaches its minimum value. The formula for \(w=(15 + 25)-AC\). Since \(AC>10\), then \(w=(40)-AC\) and \(w<30\) (when \(AC = 10\), but for non - degenerate triangle \(AC>10\), and as \(AC\) gets larger, \(w\) gets smaller. The upper - bound of \(w\) is when \(AC\) is just greater than \(10\), and the maximum value of \(w\) approaches \(40-10 = 30\))

Answer:

• How far would Pete walk if he went from A to B to C? \(40\) yards
• The direct distance from A to C is more than \(10\) yards
• The inequality \(w<30\) represents the distance, \(w\), that Pete might save by taking the direct path.