Identify the choice that is a valid second condition to determine if 3 side lengths—a

Question

, b
, and c
—will create a triangle if one condition is a<b+c
.(1 point)
Responses

a>b−c
a greater than b minus c

b>a−c
b greater than a minus c

a<b−c
a less than b minus c

a>c−b
a greater than c minus b
Remain

Answer 1

To determine if three side lengths \( a \), \( b \), and \( c \) can form a triangle, they must satisfy the triangle inequalities. The first condition you provided is \( a < b + c \).

The other triangle inequalities that must be satisfied are:

1. \( b < a + c \)
2. \( c < a + b \)

From the choices you presented, we can analyze them one by one:

1. \( a > b - c \): This does not directly correspond to a triangle inequality.
2. \( b > a - c \): Again, this does not correspond to a triangle inequality.
3. \( a < b - c \): This does not satisfy any triangle inequality as it's not valid.
4. \( a > c - b \): This resembles a triangle inequality and can be interpreted as \( a + b > c \) (rearranging gives \( a > c - b \)).

Among these options, the valid second condition to use in conjunction with \( a < b + c \) is:

\( a > c - b \)

It reflects a valid triangle inequality condition where the sum of two sides should be greater than the third when rearranging terms.