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)

Question

Responses

b>a−c
  b greater than a minus c

a<b−c
  a less than b minus c

a>b−c
a greater than b minus c

a>c−b

Answer 1

To determine if three sides \( a \), \( b \), and \( c \) can form a triangle, you can use the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.

You already have one condition: \( a < b + c \).

To find a valid second condition from the options you provided, we need to express other inequalities based on \( a \), \( b \), and \( c \). The triangle inequalities provide the following conditions:

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

Now let's analyze the given options:

1. \( b > a - c \): This can be valid based on the rearranging of the inequalities, but it is not a direct application of the triangle inequality.

2. \( a < b - c \): This does not hold true in general and cannot be derived from the triangle inequalities.

3. \( a > b - c \): This can also be rearranged but is generally not a condition arising out of the triangle inequalities.

4. \( a > c - b \): This is equivalent to \( a + b > c \), which is one of the triangle inequalities.

Among these, the valid second condition is \( a > c - b \), as it can be rearranged to reflect the triangle inequality states.

So the correct choice is:

a > c - b.