Question

Identify the choice that is a valid second condition to determine if 3 side lengths—a
, b
, and c
—will create a triangle if one condition is a<b+c
.(1 point)
Responses

a<b−c
a less than b minus c

a>b−c
a greater than b minus c

b>a−c
b greater than a minus c

a>c−b

Answer 1

To determine if three side lengths \( a \), \( b \), and \( c \) can form a triangle, the triangle inequality theorem states that the following conditions must be satisfied:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

You mentioned that one condition is \( a < b + c \) (which is indeed one of the triangle inequalities). We need to identify another valid condition.

Let’s analyze each option:

1. **\( a < b - c \)**: This does not fulfill the triangle inequality and could lead to incorrect conclusions.

2. **\( a > b - c \)**: This can be rewritten as \( a + c > b \), which is indeed a rearranged form of one of the triangle inequalities.

3. **\( b > a - c \)**: This can be rewritten as \( b + c > a \), which is also a valid triangle inequality.

4. **\( a > c - b \)**: This does not satisfy the triangle inequality either.

Both \( a > b - c \) and \( b > a - c \) are valid, but since we are required to choose one option, the best choice in line with the triangle inequality conditions is:

**Correct answer: \( a > b - c \)** (This fulfills the triangle inequality as previously explained).