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?
yards

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.

Answer 1

To find the direct distance from point A to point C, we can use the triangle inequality theorem.

Given the distances:
- \( AB = 15 \) yards
- \( BC = 25 \) yards

We don't have the distance \( AC \) directly, but we know that the distance \( AC \) can be computed or approximated if we know the position of point C relative to the line formed by points A and B.

Using the triangle inequality:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. Thus, the direct distance \( AC \) must satisfy:
- \( AC + AB > BC \) which implies \( AC + 15 > 25 \) therefore \( AC > 10 \).
- \( AC + BC > AB \) which implies \( AC + 25 > 15 \) therefore \( AC > -10 \) (not a useful boundary).
- \( AB + BC > AC \) implies \( 15 + 25 > AC \), or \( 40 > AC \) which gives \( AC < 40 \).

From the inequalities, we have:
\[
10 < AC < 40
\]

So, the direct distance from A to C is **more than 10 yards**.

Next, if Pete walks from A to B to C, the total distance would be:
\[
AB + BC = 15 + 25 = 40 \text{ yards}
\]

To find the distance \( w \) that Pete might save by taking the direct path, we can express it as:
\[
w < 40 - AC
\]
Given that \( AC \) is more than \( 10 \), the maximum saving occurs when \( AC \) is at its minimum possible value of just over \( 10 \) yards.

Plugging in \( AC \approx 10 \):
\[
w < 40 - 10 = 30
\]

This means Pete might save **less than 30 yards** by taking the direct path. Thus, the complete statements are:

- Pete would walk from A to B to C a total of **40 yards**.
- The direct distance from A to C is more than **10 yards**.
- The inequality \( w < 30 \) represents the distance that Pete might save by taking the direct path.