Using the triangle inequality theorem, choose the correct length needed for side a to be able to make this a triangle. C= 7 b= 6

Answer 1

The triangle inequality theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following conditions must hold:

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

In your case, we have the lengths \( b = 6 \) and \( c = 7 \). We need to determine the possible length for side \( a \).

Using the triangle inequality conditions:

1. \( a + b > c \)
   \( a + 6 > 7 \)
   \( a > 1 \)

2. \( a + c > b \)
   \( a + 7 > 6 \)
   \( a > -1 \) (which is always true since \( a > 1 \))

3. \( b + c > a \)
   \( 6 + 7 > a \)
   \( 13 > a \)
   \( a < 13 \)

Combining these inequalities, we get:

\[ 1 < a < 13 \]

Thus, for side \( a \) to make a triangle with sides \( b = 6 \) and \( c = 7 \), it must be greater than 1 and less than 13. Therefore, the correct range for side \( a \) is \( 1 < a < 13 \).