Question

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 \( 1 < a < 13 \).