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

Question

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