Question

A triangle has side lengths of 2 units and 8 units. Which of the following could NOT be a possible length of the third side?
7
8
9
10

Answer 1

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:

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

Substituting \( a \) and \( b \):

1. \( 2 + 8 > c \) → \( 10 > c \) → \( c < 10 \)
2. \( 2 + c > 8 \) → \( c > 6 \)
3. \( 8 + c > 2 \) → \( c > -6 \) (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**.