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
yards.
the inequality ( w < ) represents the distance, ( w ), that pete might save by taking the direct path.

Explanation:

Step1: Calculate distance from A to B to C

To find the distance from A to B to C, we add the distance from A to B and B to C.
Distance \( = 15 + 25 = 40\) yards.

Step2: Apply triangle inequality for AC

By the triangle inequality theorem, the length of any side of a triangle must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. For triangle ABC, \( |AB - BC| < AC < AB + BC \). The absolute difference of AB and BC is \( |25 - 15| = 10 \), so \( AC > 10 \) yards.

Step3: Calculate the distance saved

The distance saved \( w \) is the difference between the path A - B - C and the direct path A - C. The maximum possible distance saved would be when AC is at its minimum (approaching 10 yards), but the distance saved is \( 40 - AC \). Since \( AC > 10 \), the maximum value of \( w \) (the distance saved) approaches \( 40 - 10 = 30 \), so \( w < 30 \).

Answer:

s:

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