Using the triangle inequality theorem what is the correct length for side a if c equals 7 and b equals 6?

Answer 1

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

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

In your case, you have \( c = 7 \) and \( b = 6 \). We can use the triangle inequalities to find the range of possible values for side \( a \):

1. From \( a + b > c \): \[ a + 6 > 7 \implies a > 1 \]

2. From \( a + c > b \): \[ a + 7 > 6 \implies a > -1 \] (This condition is always satisfied since \( a > 1\).)

3. From \( b + c > a \): \[ 6 + 7 > a \implies 13 > a \implies a < 13 \]

Combining these two inequalities, we find: \[ 1 < a < 13 \]

Thus, the possible lengths for side \( a \) are any values such that \( 1 < a < 13 \).