Question

In triangle ABC, the length of side AB is 14 inches and the length of side BC is 22 inches. Which of the following could be the length of side AC?
A.
23 inches
B.
41 inches
C.
38 inches
D.
6 inches

Answer 1

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

In triangle ABC, we have:

- AB = 14 inches
- BC = 22 inches
- Let AC = x inches

According to the triangle inequality, we have three conditions:

1. \( AB + AC > BC \)
\[
14 + x > 22 \implies x > 8
\]

2. \( AB + BC > AC \)
\[
14 + 22 > x \implies 36 > x \implies x < 36
\]

3. \( AC + BC > AB \)
\[
x + 22 > 14 \implies x > -8 \quad \text{(which is always true since x is positive)}
\]

From the inequalities derived, we have:
\[
8 < x < 36
\]

Now, we can evaluate the options provided:

A. 23 inches: \( 8 < 23 < 36 \) (Valid)

B. 41 inches: \( 41 > 36 \) (Not valid)

C. 38 inches: \( 38 > 36 \) (Not valid)

D. 6 inches: \( 6 < 8 \) (Not valid)

Therefore, the only possible length for side AC from the choices provided is:

**A. 23 inches**.