Question

complete the inequality so that it represents the whole number values that side a could be to create a triangle c=7, b=6, a
<a>
options
13
1
9
7

Answer 1

To determine the possible values for side \( a \) of a triangle with sides \( b = 6 \) and \( c = 7 \), we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

The relevant inequalities for the triangle in question are:

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

Substituting the known values of \( b \) and \( c \):

1. \( a + 6 > 7 \) \(\Rightarrow a > 1\)
2. \( a + 7 > 6 \) \(\Rightarrow a > -1\) (this is always true for positive \( a \))
3. \( 6 + 7 > a \) \(\Rightarrow 13 > a\) or \( a < 13\)

From these inequalities, we can conclude:

- \( a > 1 \)
- \( a < 13 \)

Thus, the combined inequalities yield:

\[
1 < a < 13
\]

So, the whole number values that \( a \) could take are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

From the given options, the only valid value is **9**, which is a whole number within the range defined by the inequality.

Therefore, the answer is:
**9**.