Question

Complete the inequality so that it represents the whole-number values that side a could be to create a triangle.
Right side long: c=7
Left side short: a
Bottom side same length as right side: b=6

Answer 1

To determine the possible whole-number lengths for side \( a \) of a triangle given the other two sides are \( b = 6 \) and \( c = 7 \), we can use the triangle inequality theorem. This 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. \( b + c > a \)

Substituting \( b = 6 \) and \( c = 7 \) into these inequalities, we get:

1. \( a + 6 > 7 \)
2. \( a + 7 > 6 \)
3. \( 6 + 7 > a \)

Now let's solve these inequalities:

1. From \( a + 6 > 7 \):
\[
a > 7 - 6
\]
\[
a > 1
\]

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

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

Putting these together, we conclude:
\[
1 < a < 13
\]

Since \( a \) must be a whole number, the possible values for \( a \) are:
\[
a = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
\]

Thus, the completed inequality representing the whole-number values of side \( a \) is:
\[
2 \leq a \leq 12
\]