Question

In triangle ABC, the length of side AB is 15 inches and the length of side BC is 24 inches. Which of the following could be the length of side AC?
A.
44 inches
B.
41 inches
C.
31 inches
D.
7 inches

Answer 1

To determine the possible lengths of side AC in triangle ABC, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given:
- AB = 15 inches
- BC = 24 inches
- Let AC = x (the length we want to find)

The triangle inequality gives us the following inequalities:

1. \( AB + BC > AC \)
\[
15 + 24 > x \implies 39 > x \implies x < 39
\]

2. \( AB + AC > BC \)
\[
15 + x > 24 \implies x > 9
\]

3. \( BC + AC > AB \)
\[
24 + x > 15 \implies x > -9 \quad (\text{which is always true since } x > 9)
\]

From the inequalities derived, we summarize:
- \( x < 39 \)
- \( x > 9 \)

Thus, the length of side AC must satisfy:
\[
9 < x < 39
\]

Now we evaluate each of the provided options:
A. 44 inches (not valid, as \( 44 > 39 \))
B. 41 inches (not valid, as \( 41 > 39 \))
C. 31 inches (valid, as \( 9 < 31 < 39 \))
D. 7 inches (not valid, as \( 7 < 9 \))

The only possible length for side AC is:

**C. 31 inches**