Question

Complete the inequality so that it represents the whole-number values that side a could be to create a triangle.

The image shows a triangle. The top left side shows c = 7. The right side shows a. The bottom shows b = 6

___ < a < ___

Answer 1

To determine the range of values for side \( a \) in a triangle with sides \( b = 6 \) and \( c = 7 \), you can use the triangle inequality theorem, which states:

1. The sum of the lengths of any two sides must be greater than the length of the third side.

Using this theorem, we can set up the following inequalities:

1. \( a + b > c \)
- Substituting the known values:
\[ a + 6 > 7 \]
\[ a > 1 \]

2. \( a + c > b \)
- Substituting the known values:
\[ a + 7 > 6 \]
\[ a > -1 \] (This is always true for positive \( a \), so we don't need to consider it further)

3. \( b + c > a \)
- Substituting the known values:
\[ 6 + 7 > a \]
\[ 13 > a \]
\[ a < 13 \]

From the inequalities we derived, we conclude:

1. \( a > 1 \)
2. \( a < 13 \)

So, the complete inequality representing the whole-number values that side \( a \) could be is:

\[ 1 < a < 13 \]

This means \( a \) could take any whole-number value from 2 to 12. Therefore, you can fill in the blanks as follows:

**Complete inequality:**
**1 < a < 13**