To determine which lengths could NOT be a possible length of the third side of a triangle with the other two sides measuring 2 units and 8 units, 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.
Let the lengths of the sides be \( a = 2 \), \( b = 8 \), and \( c \) be the length of the third side. The triangle inequalities to consider are:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting \( a \) and \( b \):
- \( 2 + 8 > c \) → \( 10 > c \) → \( c < 10 \)
- \( 2 + c > 8 \) → \( c > 6 \)
- \( 8 + c > 2 \) → \( c > -6 \) (this inequality is always satisfied given positive side lengths)
From these inequalities, we find:
- \( c \) must be less than 10.
- \( c \) must be greater than 6.
Thus, the range for \( c \) is:
\[ 6 < c < 10 \]
Now let's evaluate the given options:
- 7: This is within the range (6 < 7 < 10).
- 8: This is also within the range (6 < 8 < 10).
- 9: This is within the range (6 < 9 < 10).
- 10: This is NOT within the range (10 is not less than 10).
Therefore, the length that could NOT be a possible length of the third side is 10.